Some helpful thoughts on church discipline from the 9MArks blog, especially the ones on sin as a result of weakness versus sin as result of rebellion.
Here’s the most important thing for you to take away from my answer: I don’t have more Bible than you have. There is no “extended cut” version that says “always do this; never do that” for most of the pastoral situations we find ourselves in.
What that means is, with just about any sin, the question of whether or not we “should” excommunicate (remove from membership and the Lord’s Table) will almost always be, “It depends.” Oh, Lord, give us wisdom.
To your scenario specifically, the first thing I’d want to know is, did he or she act more in weakness or more in open rebellion? Paul envisions different pastoral responses to different kinds of people (1 Thes. 5:14). The fact that he or she refused the counsel of the elders should weigh into your answer. But I can still imagine both scenarios, and the more a person seems to have acted in weakness, the slower I will move.
Second, how does the person talk about what he or she did. Justify it? Confess it as sin? Now, you and the member are both in the position of wanting the marriage to succeed. So don’t go fishing for a disavowal of the spouse, and always speak respectfully about the spouse. Still, your member was just given a choice between Jesus and marriage—based on God’s command to only marry in the Lord (1 Cor. 7:39)—and he or she chose marriage. Does he or she recognize that fact. Does it cause any grief?
More than that, you’re searching for any note of repentance. Christians sin. We all sin. The difference between Christians and non-Christians, however, is that Christians repent of their sin. They fight against it. Your member happened to choose a life-altering sin, so repentance doesn’t require him or her to “undo” what was done, like asking a thief to give back stolen money. But do you see any brokenness, any grief, any willingness to confess, any desire to do what’s right, any longing to hear the consolation that comes from the good news of the gospel?
Now, the person might be reluctant to say any of that because it will feel dishonoring to the spouse. You will have to help him or her separate the two things (honoring the spouse; the question of sin) by expressing an “as of this moment!” support for the spouse and the marriage. Still, that’s what you want to discern.
Assuming you believe the person is repentant, no, you should not excommunicate. But you probably need to inform the church, simply because a marriage is a matter of public record and a very visible part of a person’s life. Telling the church also gives you a chance to affirm the person’s repentance and to counsel the congregation on how to best care for the individual.
Assuming you don’t believe the person is repentant, then, yes, you might move toward excommunication.
I pray this is useful.