Not a fun situation!!!
We address this sort of situation on the front end of employment. We have both a confidentiality & a conflict of interest policy which all employees much sign off on upon hire. For conflicts of interest, they have to disclose them up front, or if a conflict arises after signing the agreement, they agree to disclose it or risk termination.
Since you're in the middle of the situation right now, that advice probably won't help you. Do you have any sort of ethics/conflict of interest/confidentiality language in your handbook or new hire paperwork? If so, that would be the first place I'd look to for assistance.
How'd you find out about the affair? Was it voluntarily disclosed? If so, I'd address the situation head-on with the individual, and remind them of their duty to the company's reputation and any fiduciary duty they may have. Then, I'd consult with your labor attorney to see whether a confidentiality/non-disclosure agreement is appropriate for the situation.
Good luck - what a sticky situation!
"You are you. Now isn't that pleasant?" -Dr. Seuss