Can a non-Catholic who married a Catholic non-sacramentally divorce without annulment?

The famous lapsed Catholic, Garry Wills, recounts how his parents had a mixed marriage and his mother, a Catholic, received the sacrament of matrimony but his father did not.

Later his father was converted to Catholicism and they went through a second ceremony in which he received the sacrament.

My question is this: before he converted, could he have divorced his wife and been free to marry again (perhaps to another Catholic, in a Catholic ceremony) without his marriage having been annulled? (Obviously, since his wife received the sacrament, this option was not available to her.)

The information is incorrect. It is impossible for a marriage to be sacramental for one spouse but not for the other. Either the marriage is sacramental or it is natural. In the case of a marriage with at least one person unbaptized it is a natural marriage. Sacramental marriage requires both spouses to be baptized.
It is also incorrect that the marriage would need to be repeated unless the first marriage was invalid, in which case it would have been invalid for both spouses. A natural marriage automatically becomes a sacramental marriage once the unbaptized spouse(s) become(s) baptized.

And finally, natural marriages are binding until death does the couple part. Thus an annulment would be needed to show no valid bond existed. Natural marriages can, in very specific circumstances, be dissolved but the situation you describe does not meet those qualifications.

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