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Hi, how long has this been going on? Do you know what is the nature of their texts ?
Are you suspecting emotional infidelity?
Men and women can be friends (in a platonic relationship). Of course, then there will be no need for secretly texting her or perhaps texting her too often.
As his wife, you do want an explanation about whether this is work related communication or personal. He had agreed not to see her by himself which may be an indication that this may be more of a codependent relationship rather than intimate. Or, she has poor boundaries and he's feeding into her.
As far as what you should do, you want to ask him to be honest and to possibly halt texting her as much unless it pertains to work, or he's helping her with something. He should set some limits with her. You are his wife and if there is nothing going on, then it would be appropriate that he honors your wish (since you feel uncomfortable) regarding this relationship.
If you're willing to give it a try, let him outline what the marital work would involve. He's shown poor judgment in having this woman with the lights off at your house. To give hi the benefit of the doubt, before making any major life changes, specifically tell him what you expect of him. Then, together come up with a plan about how to work on those issues hat had led to him getting so involved (even if it is texting/emailing) with her in the first place. What was he trying to get out of it?
Was this a codependent relationship? Depending on her age, could she be seeing him as a mentor/father figure, is he her supervisor, etc.
This may be something that outside help will be beneficial such as marital therapy or working with a relationship coach. He may have not yet apologized because he's not able to objectively see what the situation looks like.