The Catholic Church calls this situation “disparity of cult”. Here is what the Catechism of the Catholic Church (CCC) teaches about disparity of cult and mixed marriage (marriage between a Catholic and a baptized non-Catholic):
“Difference of confession between the spouses does not constitute an insurmountable obstacle for marriage, when they succeed in placing in common what they have received from their respective communities, and learn from each other the way in which each lives in fidelity to Christ. But the difficulties of mixed marriages must not be underestimated. They arise from the fact that the separation of Christians has not yet been overcome. The spouses risk experiencing the tragedy of Christian disunity even in the heart of their own home. Disparity of cult can further aggravate these difficulties. Differences about faith and the very notion of marriage, but also different religious mentalities, can become sources of tension in marriage, especially as regards the education of children. The temptation to religious indifference can then arise.
“According to the law in force in the Latin Church, a mixed marriage needs for liceity the express permission of ecclesiastical authority. In case of disparity of cult an express dispensation from this impediment is required for the validity of the marriage. This permission or dispensation presupposes that both parties know and do not exclude the essential ends and properties of marriage; and furthermore that the Catholic party confirms the obligations, which have been made known to the non-Catholic party, of preserving his or her own faith and ensuring the baptism and education of the children in the Catholic Church.
“Through ecumenical dialogue Christian communities in many regions have been able to put into effect a common pastoral practice for mixed marriages. Its task is to help such couples live out their particular situation in the light of faith, overcome the tensions between the couple’s obligations to each other and towards their ecclesial communities, and encourage the flowering of what is common to them in faith and respect for what separates them.
“In marriages with disparity of cult the Catholic spouse has a particular task: “For the unbelieving husband is consecrated through his wife, and the unbelieving wife is consecrated through her husband.” It is a great joy for the Christian spouse and for the Church if this “consecration” should lead to the free conversion of the other spouse to the Christian faith. Sincere married love, the humble and patient practice of the family virtues, and perseverance in prayer can prepare the non-believing spouse to accept the grace of conversion” (CCC 1633-1637).
The Church’s Code of Canon Law (CIC) states, “A marriage between two persons, one of whom has been baptized in the Catholic Church or received into it and has not defected from it by a formal act and the other of whom is not baptized, is invalid. A person is not to be dispensed from this impediment unless the conditions mentioned in canons 1125 and 1126 have been fulfilled” (CIC 1086 §1-2).
Canon 1125 and 1126 state, “The local ordinary can grant a permission of this kind if there is a just and reasonable cause. He is not to grant it unless the following conditions have been fulfilled:
1/ the Catholic party is to declare that he or she is prepared to remove dangers of defecting from the faith and is to make a sincere promise to do all in his or her power so that all offspring are baptized and brought up in the Catholic Church;
2/ the other party is to be informed at an appropriate time about the promises which the Catholic party is to make, in such a way that it is certain that he or she is truly aware of the promise and obligation of the Catholic party;
3/ both parties are to be instructed about the purposes and essential properties of marriage which neither of the contracting parties is to exclude.
“It is for the conference of bishops to establish the method in which these declarations and promises, which are always required, must be made and to define the manner in which they are to be established in the external forum and the non-Catholic party informed about them.”