If you want to be a model-building, research-based quant working closely with a trading desk, then you essentially need a PhD in a very quantitative subject (mathematics, physics, computer science or finance). However, I would advise very strongly against doing a PhD with the specific aim of becoming a quant at the end of it. A PhD is a lot of work and you spend a lot of time (at least three years, probably more like four) focusing on one problem. You have to be doing it because you enjoy it or you'll go nuts.
My feeling is that the more mathematical your PhD is, the better. This is because it's easy to teach economics and finance to someone who knows mathematics, but not vice versa. The other skill that's really helpful is programming ability (C++ mainly) but it's not essential.
Alternatively you could do one of the jobs which fall into the quant realm but don't require a PhD - model validation, risk management etc. The role isn't research based so you can get into it from a masters.
To be a trader you still need to know your numbers, but you can get away with a less rigorous understanding in your degree, because if you're smart enough you can learn it on your own anyway. As PB said, the maths required isn't conceptually deep, but it is important to understand it very well.