Yes, it's possible to get a graduate degree in mathematics even if your undergraduate degree is in a completely unrelated field. (Everything I say will be assuming you are in the U.S.; otherwise, I don't know.) In practice, you'll need to acquire much of the knowledge from an undergraduate degree in mathematics before starting, although you can fill in a few gaps during a master's program.
If you have unlimited time and money, then it's straightforward. You take undergraduate courses until you have completed most of a math major, either by delaying your graduation or by taking courses as a special student (i.e., not in a degree program) after your bachelor's degree. At that point, even thought you won't have a degree in mathematics, you'll be able to make a compelling case that you have equivalent background. You may not have a strong enough application to get into a top Ph.D. program directly, but you should be able to get into a master's program in a decent department, and if you do well enough there you can apply to even stronger departments for your Ph.D.
The drawback with this plan is that it's slow (you might spend two years or more taking courses before even applying to master's programs) and expensive (you'll be being charged tuition for these courses). Instead, the real question isn't whether it's possible in principle, but rather how to get to a Ph.D. program as quickly as possible, since at that point you'll no longer need to pay anything.
How efficiently you can do this depends heavily on your background and experience. If you are just starting to take college math courses, then it may take several years to prepare yourself for a master's program. If you already have a lot of experience, then you might be ready to apply this fall. I'd recommend consulting with faculty in your math department to see what they think of your background and what they would advise.
There exist certificate or post-baccalaureate programs designed to prepare people from other fields for math grad school. (See, for example, http://departments.columbian.gwu.edu/math/certificates/gradmath.) Such a program could be useful, depending on how well it fits with your background and preparation.