Unfortunately, but there really isn't a way for you to repair this and any attempt to do so will void the warranty (if your DS Lite still has one), so it would be best to send it in to Nintendo for the repair.:)

So give a call to Nintendo's 1-800-255-3700 (6 am ~ 7 pm Pacific, everyday) Customer Service number, or use their webform to contact them, and have them set up the repair process for you.

And could you edit out your name from your post, as personal information is not allowed to be revealed here, thanks.:)

Alrighty, here are your options.
1. Put a rubber band around the base of the top screen on your DSL. This is ugly, but hey, it works.
2. If your DS is still in warranty, send it in to Nintendo. OOT's contact info in his post is what you should use.
3. If it's out of warranty, you're electronic-savvy, and you don't want to spend 70+$ to have Nintendo fix it, you can find replacement shells online. I won't tell you where, though, because I don't think that's kosher on the official Nintendo forums. They should be easy to find.

Hope my post helped.

You know, my DS has the very same problem as you do. My DS has no warranty any more but I will admit, that rubber band idea is very smart if you don't want to spend any money on repairs. Hope that I helped! Reply if you want.

I never thought it would happen but my DS Lite hinge broke too!  It happened just yesterday.  I'm not gonna get it repaired though because it's way out of warranty and I never use it because I now have a DSi.  And it is still usable, the screen still stays up.

the only way i got my hinge fixed was to buy another DS lite sorry