In the case of the husband converting to Islam, there is nothing to the marriage, because in the original, sharee'ah it is permissible to marry a Muslim man with non-Muslim women (ah-lul kitab)
But in the case of opposite, that is for the woman to convert to Islam and her husband remaining a non-muslim, there are several things to consider:
The most correct of these opinions, in sharee'ah, is to divorce from him and enter into a (iddah) period. If her husband enters in islam during the (iddah) period she may return to him and does not need a new contract. If her husband enters in Islam after the (iddah) period, she may have to choose between returning to her husband or marrying another Muslim
The evidence for that in the Quran:
يَا أَيُّهَا الَّذِينَ آمَنُوا إِذَا جَاءَكُمُ الْمُؤْمِنَاتُ
مُهَاجِرَاتٍ فَامْتَحِنُوهُنَّ ۖ اللَّـهُ أَعْلَمُ بِإِيمَانِهِنَّ ۖ
فَإِنْ عَلِمْتُمُوهُنَّ مُؤْمِنَاتٍ فَلَا تَرْجِعُوهُنَّ إِلَى
الْكُفَّارِ ۖ لَا هُنَّ حِلٌّ لَّهُمْ وَلَا هُمْ يَحِلُّونَ لَهُنَّ ۖ
وَآتُوهُم مَّا أَنفَقُوا ۚ وَلَا جُنَاحَ عَلَيْكُمْ أَن تَنكِحُوهُنَّ
إِذَا آتَيْتُمُوهُنَّ أُجُورَهُنَّ ۚ وَلَا تُمْسِكُوا بِعِصَمِ
الْكَوَافِرِ وَاسْأَلُوا مَا أَنفَقْتُمْ وَلْيَسْأَلُوا مَا أَنفَقُوا
ۚ ذَٰلِكُمْ حُكْمُ اللَّـهِ ۖ يَحْكُمُ بَيْنَكُمْ ۚ وَاللَّـهُ عَلِيمٌ
حَكِيمٌ
O believers, when believing women come over to you as refugees, then
examine them. God alone is cognisant of their faith. If you find that
they are believers, do not send them back to unbelievers. They are not
lawful for them, nor are infidels lawful for believing women. Give the
unbelievers what they have spent on them. There is no sin if you marry
them provided you give their dowers to them. Do not retain your
(marriage) ties with unbelieving women. Ask for the return of what you
have spent (on them); and the unbelievers should ask for the return of
what they have spent. This is the judgement of God. He judges between
you. God is all-knowing and all-wise.
[Al-Mumtahina:10]
And from hadith:
The daughter of the Prophet (peace be upon him), Zainab, was married to Abu al-Aas ibn al-Rabee during the Jaahiliyyah (pre-islmaic period), and when she converted to Islam: the marriage between them was dissolved and she followed her father (peace be upon him), and when her husband (al-Aas ibn al-Rabee) became Muslim, the Prophet (peace be upon him) reinstated the marriage.
It should be noted, that in such cases should not prefer and give priority to worldly interests on the law of God and told him, God alone knows our interest.
