I've never been an employer.
As an employee I regularly fiddled time at one particular job. It doesn't make it right but nearly everyone did it at that place. It wasn't about getting extra time but more about leaving a bit early and other people using your clockcard. The management were aware up to a certain level in the hierarchy and turned a blind eye unless there was a purge from above. They took the view that if you want people to do long hours (occasionally 70+ hours a week and sometimes 90+) to get deadline work out of the door, then finishing a bit early if the work was all done was no big deal. That was in a different era though (up till mid/late 80s).
So I'm coming at this from a different place to some on here.
Personally, I reckon that the main problem is not about what has already happened. It's about what might happen in the future. If you let this go so long as restitution is made (either financially or with unpaid work time), then well done for being fair minded. However, other employees may take the view that they get one chance as well and to not give it, may even be classed as victimisation.
When I worked in the printing trade, there was a case of someone who got caught on a clockcard fiddle. He was sacked. However, the employer made the mistake of not prosecuting for fraud. The tribunal ruled in the sacked person's favour. Part of the reason (there were others) was that no prosecution was sought. It has to be remembered though that they were different times with different rules.
So all I'm saying is, if you sack, be prepared to back it up with weight of law. Your get out card may be the break in service as you only need to use the word "unsuitable" if he re-started and wouldn't need to seek prosecution. So, not mentioning the time fraud may be your best bet if you sack.