By an ideal conductor, I mean one with zero resistance.
The question is not about a superconductor. As I know superconductor and conductor with zero resistance are not the same thing. The question is more about what happens to magnetic field inside a wire when its resistance approaches null.
In general, it is not.
Assume a constant current flowing through a cylindrical conductor.
Applying Ampere's circuital law for a surface inside the cylinder:
$$\oint {\vec Bd\vec l = {\mu _0}\int\!\!\!\int {\vec J} d\vec s} $$ $$B2\pi r = {\mu _0}J{{\pi {r^2}} \over {\pi {a^2}}}$$ $$B = {{{\mu _0}} \over {2\pi {a^2}}}Jr$$
According to Wikipedia
In perfect conductors, the interior magnetic field must remain fixed but can have a zero or nonzero value.
Require a constant magnetic flux - the magnetic flux within the perfect conductor must be constant with time. Any external field applied to a perfect conductor will have no effect on its internal field configuration.
According to this site
The interior magnetic field is zero while the surface magnetic field is perpendicular to both the current density and the surface normal.
