Scripture teaches that divorce is only permitted for two reasons: because of adultery and because of abandonment (a non-Christian spouse leaves, or abandons, the marriage.
The main passage in Scripture that addresses the question of a marriage between a Christian and non-Christian is 1 Corinthians 7:12-16: “To the rest I say (I, not the Lord) that if any brother has a wife who is an unbeliever, and she consents to live with him, he should not divorce her. If any woman has a husband who is an unbeliever, and he consents to live with her, she should not divorce him. For the unbelieving husband is made holy because of his wife, and the unbelieving wife is made holy because of her husband. Otherwise your children would be unclean, but as it is, they are holy. But if the unbelieving partner separates, let it be so. In such cases the brother or sister is not enslaved. God has called you to peace. For how do you know, wife, whether you will save your husband? Or how do you know, husband, whether you will save your wife?” (ESV)
In 1 Corinthians 7 Paul is addressing various issues of marriage, singleness, divorce and re-marriage. In verses 12-16 he addresses the question of marriages where a believer is married to a non-Christian. Paul’s basic position is given in verses 12-13: that Christians in marriages with non-Christians should not divorce. Just a comment on the start of verse 12: “To the rest I say (I, not the Lord)” – this probably refers to the fact that Jesus didn’t address this issue in his ministry, therefore it is Paul speaking rather than reiterating Jesus’ teaching. However, since Paul is an apostle, his words have the same authority as Scripture and therefore as Jesus himself (cf. 2 Peter 3:15-16).)
In verse 15, Paul then addresses the question of what should happen if the unbeliever in the marriage issues a divorce. In this case the divorce is permitted, and the Christian can re-marry if they so desire (v. 15).
Paul’s reason for a Christian not divorcing a non-Christian is presumably based on Jesus’ own teaching that in marriage that the two have become one flesh, and therefore shouldn’t be separated (Matthew 19:5-6). Additionally, the Christian in such a marriage doesn’t know what will happen: maybe their non-believing spouse will, in time, become a believer (v. 16).