The first question to consider, is can it truly remain an issue outside of work? By that I mean, does it affect either your, or your employee's ability to do their job, to work with their co-workers and will it affect employee behaviour at work?
The second question to consider, is what are the likely outcomes of this affair and its link to your marriage breaking up on the office relationships if it becomes public? Will it have a wider affect on other staff?
The third question to consider, is can you continue to do your job whilst working with this individual?
Based on what you have said in the question, I believe it quite likely that your ability to manage this employee has been subtly compromised. You have an emotional response to him, and probably do not trust him either. The fact that he expects that this would not have any effect on his ability to work with you shows a lack of understanding of how people work.
You could terminate his employment contract for his conduct. The employee has acted in a way that it is reasonable to expect that you would be uncomfortable to continue to work with him, he has also undermined the trust you had in him. Both are arguable grounds for immediate termination with no compensation. The cost here is both the internal shock in the company that this person has left with no explanation, the loss of internal knowledge, and the risk that they attempt to claim compensation and the distraction on the business that can cause.
You could continue to employ him, and see how it plays out. You will be operating sub optimally, that over time this employee will not be able to carry out their duties properly - his loyalties are bound to be divided between his work and his home, and the overall cost on lost opportunity in terms of your focus and delivery, and this employee holding back on idea generation for personal reasons will have a significant cost to the business.
You could try to pay him to leave. Based on that he believes he could continue to work with you, I suspect that this would be an expensive option. But it's worth finding out what the cost would be.
In your situation, I would sack him, with a months pay, and a glowing reference. If he does go to court - and wins, take solace that every 2 pennies you pay in lawyers fees is one less penny that your wife will get in the divorce settlement.