I can affirm what a few people have said about the entrance exams and such being very difficult to get into any university in Spain. After I graduated high school, I spent a year studying abroad in Madrid through the Rotary Youth Exchange program. They placed me in COU (Curso De Orientación Universitaria). The classes I took were in preparation for this entrance exam, Selectividad (I think that is the exam everyone else is referring to). Of course I didn't have to take it, as I was there really just experiencing and learning the language, but yes I could tell it would be very thorough. I think I was taking like eight classes, and you chose your emphasis - Letras (philosophy, history, etc) or Ciencias (chemistry, more math, etc). Some don't pass and have to take that year all over again preparing for the next year. And what score you get determines what you can study.
Anyway, about getting your degree abroad. You may look into Saint Louis University - they have a campus in Madrid and are accredited by both Spain and the US (in other words, your degree would be valid in both countries). I was looking into that to do my Master's there, which you can do in four summers. Here is their website:
http://spain.slu.edu/ It doesn't look like you can get your entire undergrad degree at this particular university - but at least you could study there for a year, and know what you are taking validates in both the US and Spain. It's not easy to find a school like that.
Hope this helps!
UPDATE: I am so sorry ya'll - just noticed a couple of people already mentioned SLU. My bad!