He married her after she became Muslim, but she does not want to get divorced legally from her non-Muslim husband
Question

I met a woman who had recently embraced Islam. She was married to a non-Muslim man, but they had separated several years previously, without divorcing. I know that the Islamic ruling in this case is that as soon as a woman embraces Islam, intimacy between them becomes prohibited. Because I was aware of this ruling, I married her according to Islamic sharia, and we have a child now. Is this marriage valid? I asked her to seek a divorce from her previous husband, but she refused. She tells me that she wants to remain without a legal divorce from him in order to benefit financially from that, because they paid taxes together and they get a monthly allowance for their children (child benefit). She also says that she will never do that, because she does not trust me completely. My questions are: 1. Is this marriage of mine regarded as valid? Is my marriage to a woman who, according to the man-made laws of the land is still married to someone else, permissible according to Islamic teachings? 2. Do I have to leave this wife of mine who does not want to marry me legally [according to the law of the land], even though – ma sha Allah – she tries hard to adhere to the teachings of Islam? 3. What should I do about my child who I have from her? In other words, if my marriage to her is not valid, does this child have any rights over me and do I have any rights over him?
Praise be to Allah.Firstly:
If a woman becomes Muslim and her husband remains a disbeliever, intimacy between them becomes unlawful and she becomes completely divorced from him at the end of her `iddah, which starts from the time when she became Muslim. If her husband becomes Muslim before the end of her `iddah, the marriage remains as it was, but if he does not become Muslim before her `iddah ends, the marriage is deemed to be annulled from the time when she became Muslim.
Ibn Qudamah (may Allah have mercy on him) said in al-Mughni, 7/117: If one of them became Muslim after consummation of the marriage, there are two views which were narrated from Ahmad, one of which is that it depends on the end of the ‘iddah. If the other spouse becomes Muslim before it ends, then they are still married, but if he does not become Muslim before the `iddah is over, they are regarded as having separated from the time she became Muslim, so there is no need to start counting the `iddah all over again. This is the view of az-Zuhri, al-Layth, al-Hasan ibn Salih, al-Awza`i, ash-Shafa`i and Is-haq. A similar view was narrated from Mujahid, `Abdullah ibn `Umar and Muhammad ibn al-Hasan. End quote.
The ‘iddah is three menstrual cycles for a woman who menstruates, or three months for one who does not menstruate.
Based on that, if you married her after the end of her `iddah, then your marriage is valid, and it does not matter if she is still married to someone else legally (according to the law of the land).
But if your marriage took place before her `iddah ended, then the marriage is not valid, and she must complete her `iddah, then you may marry her after that.
Secondly:
Despite your marriage being valid – If it was done after the `iddah – she must still get out of her previous legal marriage for several reasons:

Because several rulings result from her still being married legally, such as her inheriting from that man or him inheriting from her; they cannot inherit from one another according to Islamic law.
If the law of the land does not give the allowance to a divorced woman, it is not permissible for her to use tricks to take it. If the law dictates that she should pay money or taxes so that she will get something in return as benefits for her and her children, she has the right to try to get what she paid only, then she should refrain from taking it.
You must register your marriage to her in order to protect your rights and hers, and those of your children. That can only be done by her getting divorced from her former husband.

As for her fear that you will divorce her, we do not think that she wants to go back to her former husband, because he is a disbeliever and is not permissible for her to back to him, and because if he becomes Muslim, it is not permissible for her to go back to him except with a new marriage contract. What appears to be the case is that she wants to keep receiving the money that the state gives to her and her children. In that case, she should be told: By keeping your first marriage contract, you are getting involved in many bad things and you may be taking something to which you are not entitled. If, because of that, you are not registering your marriage, then you are neglecting a religious obligation.
Thirdly:
The child that has been born from your marriage is to be attributed to you, because if the marriage took place after the `iddah ended, it is valid, as noted above. If it took place before the `iddah ended, then it is an invalid marriage, but you believed it to be valid, so the child is to be attributed to you.
You must handle this matter with gentleness and wisdom.
And Allah knows best.