Michelle has pointed out that there's a distinction between a sacramental marriage and a valid marriage. I read a little more about it, but I'm not going to explain here what I've learned, because I'm sure to make mistakes again and I'll really confuse everyone. Instead I'll give a link to the Question and Answer page about marriage on the US Conference of Catholic Bishops website. Hopefully that will be helpful to anyone who is interested in this.
Here is another article that talks about why mixed marriages are not recommended, even though they are sometimes permitted.