Light is bent near a mass (for example when passing close to the sun as demonstrated in the famous sun eclipse of 1919). I interpret this as an effect of gravity on the light.
However, it seems (to me, at least) that light is not accelerated when it travels directly toward the (bary-)center of the sun. The same gravitational force applies yet the speed of light remains constant (viz. $c$).
What am I missing?
You missed a key aspect of general relativity (GR):
In general relativity, the presence of mass and energy warps four-dimensional space time, thereby inducing spatial curvature. The greater the presence of mass and energy in a given location, the greater the induced spatial curvature. When any particle (massless or not) travels into this curved space, the particle will continue to travel in a straight line (absent outside forces); but, since the space it is travelling upon is curved, its global path will be curved.
As an analogy, draw two straight lines on a sphere (a curved surface) travelling in different directions. Locally (at small two-dimensional distances), the lines travel in a straight direction, without veering. Globally (at three dimensions), we see that its path is curved and will inevitably intersect (on the other side of the sphere). We call these paths geodesics. The math concerning geodesics involves differential geometry which makes heavy use of multivariate calculus.
Now back to general relativity. GR predicts that the gravitational forces we observe are the manifestation of four-dimensional spacetime being warped by the presence of mass-energy. A common analogy made is the trampoline-well model shown below. A heavy mass sitting upon a trampoline curves the surface of the trampoline. Any objects then move towards the heavy mass has its path deflected towards it. Now, I must stress an important simplification made in such diagrams: these diagrams reduce four-dimensional spacetime to three spatial dimensions. The XY plane of the diagram represents the XYZ components of spacetime whereas the Z axis of the diagram represents the T component of spacetime. For math lovers, they are reducing $(\vec{X}, \vec{Y}, \vec{Z}, \vec{T})$ to $(\sqrt{\vec{X}^2 + \vec{Y}^2}, \vec{Z}, \vec{T})$
Now here's the cool part:
Now instead of its path curving along the $\vec{X}\vec{Y}$ plane as seen in the photo above, its path would curve against $\vec{Z}$ (vertical). In this context though, $\vec{Z}$ does not refer to the Z direction, but to $\vec{T}$. What this means is that, observers would see the particle 'accelerate' in through time and apparently 'slow down'. Namely, they will see gravitational time dilation.
EDIT. I made a mistake: the light does accelerate! It merely does so according to the rules of special relativity.* When objects (massless or not) pass near a gravitational well they pick up gravitational energy and accelerate, thereby gaining kinetic energy. For objects with mass, this means an increased velocity (hence gravitational sling shots). For massless particles (such as photons), this typically means an increased frequency or blue-shifting as Jeremy pointed out in a separate answer. (Thank you Peter, Rob, and Jeremy for pointing out this oversight.)
You may have noticed a contradiction here. According to special relativity and observations, objects in gravitational wells do accelerate. Case in point: gravitational sling-shooting. According to general relativity though, the 'gravitational force' we observe is a manifestation of four-dimensional space-time curvature. So which is it: is there a force or not? Not-really: it's a matter of frames-of-reference. From our reference frame we see acceleration; but, from the four-dimensional space-time reference we see pure geodesic motion.
One thing that the previous answers are missing -- the light is accelerated; it just is accelerated according to the rules of special relativity, which says that it cannot pick up speed when already travelling at the speed of light.
Instead, it gains kinetic energy the way a photon gains kinetic energy -- by being blueshifted to a higher frequency, which does translate to more energy, according to the Planck relation $E = h\nu$.
Currently, there is no evidence that photons have mass, and it is generally accepted that they are massless particles.
Nonetheless, gravitation does affect the path of photons, because the bending of space-time causes all particles to travel on curved paths, including massless ones. But that does not mean than light will be accelerated. The speed of light (299,792,458 m/s) is an absolute maximum, and it may not decrease from that nor may it increase.
Does the speed of light change due to the Sun? Well, yes and no. There are two ways in which one can think about speed in General Relativity. One is the coordinate speed, which means the rate of change of the spatial coordinate with respect to time coordinate of the coordinate system that you can choose at your will. Another is the speed as viewed from a special frame, namely the locally inertial frame in the vicinity of the considered light quanta. The fundamental aspect of General Relativity is that the Physics in a locally inertial frame is exactly the same as the Physics of Special Relativity. But due to gravity, these small-small local inertial frames are so arranged globally that a global inertial frame can't be formed. Now, in a generic frame of reference, i.e., in a generic coordinate system, the speed of light can certainly be different from $c$ and, in fact, it can even change with time.
For example, the speed of a photon moving radially in the vicinity of a spherically symmetric and static object is given by $v=\dfrac{dr}{dt}=c\bigg(1-\dfrac{2GM}{rc^2}\bigg)$ if you choose your spatial coordinates to be the spherical coordinates centered at the massive object with radial coordinate $r$ and keep track of time with a clock situated far away from the spherical object (star). As you can clearly see, the speed can vary with the radius $r$. Although, the speeding up and speeding down occurs in somewhat counter-intuitive fashion. An out-going photon seems to be speeding up and an in-going photon seems to be slowing down. Again, if you go to a local inertial frame, the speed is invariable $c$ but you don't have any frame that is inertial and can describe the motion of light for a finite time or within a finite region of space.
You are missing special relativity and general relativity. In special relativity the speed of light in vacuum is always c, no matter the reference frame of measurement.
Also classical electromagnetism, light, emerges from a confluence of the quantum mechanical constituents which are photons and have zero mass. A photon aiming at the barycenter of the sun is attracted by the gravitational field of the sun but the effect is not a change in velocity , but on its energy which is $E=h\nu$ and therefore the extra energy increases the frequency while the velocity remains at $c$.
In Einstein's 1911 paper "The influence of gravitation on the propagation of light", he pointed out that from his generalizrd theory of relativity, the speed of light as viewed from us is different at different locations in a gravitational field. From the Huygens principle, a change in the speed of light causes the wave front to tilt over toward the sun. The exact amount of the deflected angle of star light passing near to the sun is by his calculation 0.85 arcsecond(which he later corrected it to be 1.75 arcsecond). The bending of light path near to the sun is thus caused by the variable speed of light.
