Sheikh Ibn Baz, a prominent Muslim scholar, states,

Adultery is one of the most serious of impermissible actions and the worst of major sins. Allah warns the disbelievers, murderers and adulterers of multiple punishments on the Day of Resurrection, as Allah says: 

(And those who invoke not any other ilah (god) along with Allah, nor kill such per
son as Allah has forbidden, except for just cause, nor commit illegal sexual intercourse — and whoever does this shall receive the punishment. The torment will be doubled to him on the Day of Resurrection, and he will abide therein in disgrace. Except those who repent and believe (in Islamic Monotheism), and do righteous deeds.
” (Al-Furqan 25:68-70) 

The one who has fallen into such sin has to repent to Allah sincerely, and follow that with sincere belief and righteous deeds. Repentance is sincere if the penitent gives up the sin, regrets what has happened in the past and resolves never to go back to it, out of fear of Allah and awe of Him, hoping for His reward and fearing His punishment. Allah says: 

And verily, I am indeed forgiving to him who repents, believes (in My Oneness, and associates none in worship with Me) and does righteous good deeds, and then remains constant in doing them (till his death)” (Ta-Ha 20:82)

Every Muslim man and Muslim woman must beware of this great evil and the things that lead to it, and hasten to repent from whatever is already past. Allah will accept the repentance of those who are sincere and forgive them.” 

Regarding your question, we quote a fatwa of Sheikh Ahmad Kutty, a senior lecturer and an Islamic scholar at the Islamic Institute of Toronto,Ontario, Canada, on this concern who states:

“If a person has been guilty of a sin and Allah has covered him or her, then he or she should not uncover the veil that Allah has placed over her sin. Rather, they should ask forgiveness of Him and repent to Him sincerely, for He is, above all, all-Compassionate, All-Merciful, All-Forgiving. Allah says, “I am most assuredly Most Forgiving to those who repent, cherish faith and do good works.” (Ta-ha: 82)
It is wrong to say that sin of adultery is expiated and forgiven only if the person were stoned to death or given hundred lashes. In fact, such punishments are extreme measures that can only be applied in rare cases.

You must therefore take it to your heart that Allah’s forgiveness is not at all conditional on submitting yourself to such punishments. You can surely expect Allah’s forgiveness if you are willing to take the necessary steps towards true and sincere repentance: Among these steps are: Feeling deep remorse for your sin, refraining from the sin as well as from all the leads and circumstances that led you to the same in the first place and resolving firmly in your mind never to do it again. Finally such a determination should be followed by whatever good works you can do to expiate for your sin.
allah says, “Verily good deeds wipe out bad deeds.” (Hud: 114). The Prophet (peace and blessings be upon him) said, “Allah does not wash out bad deeds through bad deeds; rather He washes bad deeds only through good deeds.”
So repent sincerely and get yourself occupied with good works to the best of your ability. Once you have followed these steps your sin is blotted out. The Prophet (peace and blessings be upon him) said, “A person who has repented (sincerely) is like one who has never sinned.”
excerpted, with slight modifications, from:

As regards your marriage to the one with whom you committed this sin, this is only permissible if both of you have repented sincerely to Allah. After attaining repentance, you may consider the matter of your marriage and if you find it a suitable matter, you may then go on in it.

May Allah guide all of us to what pleases Him!