As Jérôme Renard said, this is really a Python vs Ruby question, not Django vs Rails. And, as can be gathered from the other answers to this question, there is no clear winner. Both are good, modern, well-designed, flexible, good for your productivity (unlike PHP which fails on all these counts). So choosing between the two really boils down to a fit between your personality/preferences and the language strengths.

These are the factors to consider:

• Python really believes that code readability is the most important thing. Hence there is one-true way of writing code. Ruby believes in giving programmer the freedom and power. These are two different approaches to programming, and which you choose depends upon your personality. See these two questions for more on this tradeoff: What are the advantages of Ruby over Python? and Python seems to be a preferred teaching language in many top universities, along with C/C++, Java, and Scheme. What is it about Python that makes it more attractive? Why is Python preferred to Ruby in academia? (I ended up choosing Python because I'm at a stage where I prefer readability over flexibility)
• Are you going to stick to web development mainly, or will you be branching out into other areas. I think Ruby+Rails+etc holds a slight edge over Python+Django+etc for web dev, and has more mindshare, whereas for things like data manipulation, analytics, system administration, scientific programming, Python is stronger. (I believe that over the next few years, data analysis is going to be big, and web developers cannot / should not ignore it - so Python wins again for me)
• Job prospects: this is a tricky one. I think the TIOBE index (http://www.tiobe.com/index.php/content/paperinfo/tpci/index.html) clearly shows Python as being way ahead of Ruby in jobs. However, this point is related to the previous point. I would expect that a lot of the python jobs are non-web-dev. So, assuming you're interested in web development, I would guess that there are more Ruby jobs right now.
• Update: Heroku now supports Python/Django, so this bullet point is no longer entirely relevant. -Thanks Quora User and Carl Gao. (Kept here for historical purposes: "Hosting" - this is more of a short-term thing, but I'm listing it for completeness - Rails has Heroku, the very easy and convenient platform to deploy Rails applications. Python/Django does not have a Heroku equivalent (although a bunch are in the works - What is the Heroku equivalent for Django applications?). Python does have Google AppEngine, but it not entirely ideal for a number of reasons. In any case, I don't think this aspect will continue to be an issue for more than 6-9 months.)

And I disagree with Nicolas Perriault (and agree with Jean-Hadrien Chabran) that you cannot really learn both in a reasonable amount of time. Sure, you can scratch the surface of both, but you don't really get a real feel for the framework unless you've been writing non-trivial code for at least 6 months (each). You will need to take a call based on your gut feel.