When is the marriage covenant broken? When a married person actually commits the sexual act outside of marriage or when his spouse divorces him?
Covenants are vows that last for the lifetime of the parties entering into the covenant. "For the married woman is bound by law to her husband while he is living; but if her husband dies, she is released from the law concerning the husband" (Romans 7:2). Here, "law" is being used as a synonym for "covenant."
The question, then, is what is meant by "broken." Do you mean that the terms of the covenant were violated or that the covenant is no longer in place?
When was the Law of Moses broken? When the Israelites sinned (Isaiah 59:1-2) or when God rejected them (Isaiah 50:1)? The point in Isaiah 50:1 is that God did not formally divorce Israel and so could call them back to Himself, but the problem that caused the separation had to be settled first. "'Behold, days are coming,' declares the LORD, 'when I will make a new covenant with the house of Israel and with the house of Judah, not like the covenant which I made with their fathers in the day I took them by the hand to bring them out of the land of Egypt, My covenant which they broke, although I was a husband to them,' declares the LORD" (Isaiah 31:31-32). The sins of Israel broke the terms of the covenant. As a result, God declared He would replace the covenant with a different one. But that implies that until the covenant was replaced, the old covenant remained in place. It was under the terms of that old covenant that God punished Israel for her sins by sending them into exile.
Adultery breaks the terms of the marriage covenant between a husband and wife. But an act of adultery does not remove the covenant. People cannot nullify a covenant put in place by God. "So they are no longer two, but one flesh. What therefore God has joined together, let no man separate" (Matthew 19:6). Later, Jesus stated that if a person commits adultery and is divorced, the other spouse could enter into another marriage covenant, but the adulterer could not because he is still bound by the covenant he entered into (Matthew 19:9). This is the only allowance for releasing someone from a marriage covenant that God set. Even if a couple divorces for other reasons, they have to remain unmarried until such time that they reconcile their differences (I Corinthians 7:10-11). Thus, neither adultery or divorce completely end a marriage covenant.