I was in a very similiar position. My undergrad was not in psych, I had been out of school for quite some time, my GRE scores were very high (I think this can go a long way... although the research shows it shouldn't), and to be quite honest I had no psychology research experience; in fact no research experience period (although I really love research now). I had 4 psychology classes under my belt when I applied to grad school and was actually given offers by every school I applied to (although, only one was clinical).
Yet, here I am in my final (well pre-internship) year in a well respected Clin Psych Ph.D. program. I think it's important to be a thinker and to find a good match with a faculty member. You've always got a chance. Emphasize your strengths, be outgoing, let them see who you are. If they don't like it then you wouldn't have been happy there anyway.
Are you primarily interested in research? If so, why not do something where your science/math/stats background would be put to better use (e.g., Quantitative Psychology/ Psychometrician)?