Can a couple that is living together be baptized into the Mormon church?
A couple that is living together without being married will need to live in separate locations, or make some other arrangement. The principle behind the rule is that sexual relations outside of marriage are contrary to the gospel of Jesus Christ.
This can be a very difficult decision, and I know of several couples who have made this decision before being baptized. I know there are others who decided to continue living together and not be baptized. It's a personal decision, and of course they can still attend church services even if they decide not to be baptized.
The LDS Church teaches that chastity before marriage is an essential part of living the gospel of Jesus Christ. However, just as Jesus did, the Church gives members and potential members every opportunity to change their lifestyle without condemning them.