Just go into receiving those bill assuming you will see something you don't want to. I find that's the best way to go into those situations...then, if it's not there you are pleasantly surprised. However, with him making a comment like that...I'm betting he's been in contact with the OW. Just assume he has.
IF he has...THEN, he needs to change his cell phone number. Obviously after that you will have all bills for that phone available to you...if her number appears again you'll know he had to have given it to her. If there are other suspicious numbers you can deal with that then.
This is of course IF he has been in contact with her....or even someone new.