A The current policy of the Catholic Church is this: The Catholic partner must agree to two statements:
"I reaffirm my faith in Jesus Christ and intend to continue living that faith in the Catholic Church."
"I promise to do all in my power to share my faith with our children by having them baptized and raised as Catholics."
The non-Catholic partner does not have to promise anything, but the priest arranging the marriage must certify that the non-Catholic is aware of the commitment that the Catholic spouse has made.
This policy comports with the church's Code of Canon Law, as it was revised in 1983, and it represents a notable change in wording from the earlier Code of 1917, which required both parties to sign written promises that their children would be baptized and brought up as Catholics.
If the non-Catholic partner simply refuses and insists that the children will not be raised Catholic, a diocese could still grant permission for the marriage so long as the Catholic party agrees to do whatever he or she reasonably can, within the context of the marriage, to have the children be Catholic.
Certainly the church's primary goal is to ensure the survival and stability of the marriage itself; there are situations where the wife, for example, continues to attend Mass every Sunday and would dearly love to pass her faith on to her children, but has conceded that, for the sake of peace in the family, she cannot insist on this over her husband's strong objection.
In such a setting, though, one might be inclined to ask whether the marriage itself was a wise and prudent one.
Religious beliefs are (hopefully) at the core of who a person is, and a wide gap might be hard to bridge.
At the same time, I have seen ecumenical or interfaith marriages work quite harmoniously, with deep regard for each other's beliefs and the transmission of solid faith to the children.
What is essential is that the spouses, well before the marriage, examine the religious issues deeply and with a clear understanding of each other's point of view.
It troubles me to find, as I sometimes do, that a couple already engaged to be married has not yet even discussed such a fundamental topic.