there is no rule barring felons from serving in Congress

although, also, it is not a requirement.

The Senate's longest-serving Republican, Stevens said he had no idea he was getting freebies. He said he paid $160,000 for the project and said he believed that covered everything.
Stevens asked for an unusually speedy trial, hoping he'd be exonerated in time to return to Alaska and win re-election. He kept his campaign going and gave no indication that he had a contingency plan in case of conviction.

No one anticipated that the levies would fail....