Praise be to Allah.
What happened between that man and that woman – based on what is mentioned in the question – is undoubtedly zina. Where was the woman’s wali (guardian) in this marriage? It is strange that the angels were witnesses to this marriage but they did not see those witnesses.
What you have to do is repent to Allaah from what you have done, and realize that you have committed a major sin. The Prophet (peace and blessings of Allaah be upon him) said: “No adulterer is a believer at the time when he is committing adultery.” Narrated by al-Bukhaari (2343) and Muslim (57). For more information please see question no. 11195 and 21223.
If a person repents, Allaah rejoices at his repentance and accepts it and forgives him, as Allaah says (interpretation of the meaning):
“And those who invoke not any other ilaah (god) along with Allaah, nor kill such person as Allaah has forbidden, except for just cause, nor commit illegal sexual intercourse and whoever does this shall receive the punishment.
69. The torment will be doubled to him on the Day of Resurrection, and he will abide therein in disgrace;
70. Except those who repent and believe (in Islamic Monotheism), and do righteous deeds; for those, Allaah will change their sins into good deeds, and Allaah is Oft‑Forgiving, Most Merciful”
See also questions no. 27113, 624 and 20949.
They have to cut off all ties with one another, so as to close the door to temptation and desire. If they want to have a good, legitimate relationship, then they should enter houses by the proper doors, as Allaah has prescribed, and they have to fear Allaah and be patient in heeding His sacred limits. A person may be able to put up with the pain of separation but he will not be able to put up with the heat of the Fire. So let them keep away from one another, so that they will not fall into doing that which will earn the wrath and punishment of Allaah, or let them hasten to get married in the proper Islamic manner, after repenting from what they have fallen into of zina. And Allaah knows best.