Islam permits the husband and wife who are incapable of having a baby through normal fertilization to resort to In-Vitro Fertilization with certain conditions and regulations. However, the wife is not allowed, as far as the question in point is concerned, to resort to using her husband’s frozen sperm after his death or after divorce.
The prominent Muslim scholar, Sheikh `Abdul-Khaleq Hassan Ash-Shareef, states the following:
It is not permissible for the woman to use the frozen sperm of her husband after divorce or after his death. The issue is permissible only when both the husband and wife are living together (i.e. they are not divorced) and both of them are alive.
Hence, it is not permissible for the ex-wife or the widow to use her husband’s frozen sperm after his death or after divorce. The woman may face a very difficult situation as regards this, especially when she gets pregnant, while her husband is dead or she is divorced.
In addition, the Fatwa issued by the European Council for Fatwa and Research states:
It is permissible for the wife to use the sperm of her husband for fertilization unless she is divorced, or the husband dies.
In case the wife is separated from her husband (i.e. by divorce or death), it’s permissible for her then to get rid of the frozen sperm or its remnants.