The answer is probably no; full disk encryption (at least, using the native dm-crypt method) officially requires completely wiping the Ubuntu partition and reinstalling it inside an encrypted container. It is not possible to do this incrementally without employing third-party tools that come with a lot of disclaimers about the risks involved in this, e.g. LUKS In-Place
However, encrypting the whole disk tends to be unnecessary, since there is little security implication in having the standard parts of an Ubuntu system stored unencrypted. You may want to look at either encrypting your home directory (which should be possible without reinstalling, even if not easy), or just creating a single encrypted folder for sensitive files.