She married an American man who announced his Islam, then he stopped praying and denied that Allah created the heavens. What should she do?

Praise be to Allah.

Firstly: 

We ask Allah to make things easy for you and to give you a way out of hardship and relief from distress. 

Secondly: 

It is not permissible for a Muslim woman to remain married to a kaafir, because Allah says (interpretation of the meaning):

“O you who believe! When believing women come to you as emigrants, examine them; Allaah knows best as to their Faith, then if you ascertain that they are true believers send them not back to the disbelievers. They are not lawful (wives) for the disbelievers nor are the disbelievers lawful (husbands) for them”

[al-Mumtahanah 60:10]. 

Although the scholars differed as to whether the one who does not pray is a kaafir or not, they did not differ as to whether the one who denies that Allah created the heavens and the earth is a kaafir. 

If a Muslim apostatises, his marriage to his Muslim wife is rendered null and void. If he comes back to Islam, then their marriage is reinstated, but if her ‘iddah ends and he is still insisting on apostasy, then she is now in charge of her own affairs and may marry someone else if she wishes. Based on that, if your husband persists in his ways, then the marriage between you is annulled according to sharee’ah and you are no longer his wife. 

You have to strive to get out of this marriage by whatever means, either by talaaq or khula’ or something else. 

Beware of making the children a barrier between you and leaving this man. 

Shaykh Muhammad ibn Saalih al-‘Uthaymeen (may Allah have mercy on him) said: 

Many women -- Allah forbid -- are prevented by the presence of children from seeking annulment of marriage. This is a serious matter. It should be said: Annul the marriage; it is not permissible you to stay with this kaafir who does not pray. Your children will never leave you so long as their father is like this. He has no guardianship over them, because the kaafir can have guardianship over a believer. “And never will Allaah grant to the disbelievers a way (to triumph) over the believers” [al-Nisa’ 4:141]. So you and your children cannot be separated. As for this husband, there is nothing good in him. You should forsake this kaafir husband and it is haraam to let him be intimate with you; this is a great evil. End quote. 

Al-Sharh al-Mumti’ ‘ala Zaad al-Mustaqni’, 12/250 

We advise you to take your case to the Islamic centres in your country and to the Muslims, and ask them for help. They know better than us how to deal with this problem, as they are present in that country. 

We criticise your family's attitude towards you. We understand from your question that you chose this husband and chose to live with him without your family's approval. This was a great mistake on your part, and you have paid the price for that, but that does not mean that your family should abandon you. Keep getting in touch with them and ask good and wise people among your relatives to mediate between you and them so that they will help you and stand beside you during this difficult time. 

You should understand that what is happening to you now is the result of previous mistakes, among the greatest of which were getting married and travelling without your family's approval, then choosing to reside in a kaafir state rather than living in a Muslim land. 

This is the price for living in that doomed land. 

You have to try to leave that country without delay, and go back to your family and your country; take the children with you and bring them up in Islam, teach them to memorise the Book of Allah, instead of them seeing the bad example that prevents them from following the true path. 

Hence it is right that you should weep night and day for how you have ended up. But weeping does not achieve any benefit. So hasten to do something and strive to change this situation, by annulling the marriage contract and not letting him be intimate with you, and by returning to your family and your country. 

We ask Allah to make things easy for you and to protect your religious commitment and to make your children righteous. 

And Allah knows best.