Yes, undergraduates frequently prove new things, in the sense that every year there are new, publishable results proved by undergraduates. So, although a relatively small number of undergraduate math students participate in true "research", there are certainly students who are able to make nontrivial discoveries as undergraduates, and more than one might initially think. I have been at prestigious research schools and at anti-prestigious regional universites in the U.S.A. At every school I have been, there were undergraduates in mathematics with the aptitude for publishable research. The talent needed may not be "common", but it is certainly not "rare". The obstacles are primarily cultural, not intellectual.
The topic of undergraduate research has also been the subject of a question on MathOverflow, which makes for good reading.
For an example from personal experience: I recently published a peer-reviewed paper in what I consider to be a high-quality journal (and which is not in any way a "student" journal), with an undergraduate student co-author, who discovered the proof of one of the main theorems on his own between two of our research meetings.
Another example is the journal Involve, which is devoted to genuine student research. From their self-description:
Involve showcases and encourages high-quality mathematical research involving students from all academic levels. The editorial board consists of mathematical scientists committed to nurturing student participation in research.
Submissions in all mathematical areas are encouraged. All manuscripts accepted for publication in Involve are considered publishable in quality journals in their respective fields, and include a minimum of one-third student authorship. Submissions should include substantial faculty input; faculty co-authorship is strongly encouraged. In most cases, the submission (and accompanying cover letter) should come from a faculty member.
Involve, bridging the gap between the extremes of purely undergraduate-research journals and mainstream research journals, provides a venue to mathematicians wishing to encourage the creative involvement of students.
One thing that undergraduates are unlikely to have is the breadth of knowledge that is expected for PhD recipients. Particularly in mathematics, PhD students are examined in a range of subjects, and are expected to have mastered large parts of the undergraduate curriculum. Undergraduate research often involves learning enough about one particular area to prove new theorems. The student still needs to spend time learning other areas to have the knowledge expected of a PhD.
The real key for undergraduates who are looking to do publishable research is to find a collaboration with a good faculty mentor. Independent research by undergraduates is indeed quite rare (in fact, the majority of mathematics papers currently published have two or more authors - even experts benefit from collaboration). The MathOverflow thread linked above has more advice from other mathematicians.