Heyho.

A few other suggestions, since I had a similar problem with my HD5770 when one of my fans died:
1. Set a framerate cap, less work means less heat.
2. Lower the core clock and voltage of the card, less power means less efficiency, but also less heat.

For setting a framerate cap I use RadeonPro myself: http://www.radeonpro.info
For voltage monitoring MSI Afterburner: http://event.msi.com/vga/afterburner/download.htm

Midoryu

Edit 3, 3 hours later:
It's "framerate cap", not "frame cap". Duh...
Typing error. Underlines. Italic has it's issues.

The thermal paste commonly used for CPU is not advised for GPU, beware.
As I said 'Thermal pads' (google that) do the job nicely (because there is the gddr to also take care of).

But as others said, your priority should be to get a low-end backup gpu (or do you have an integated vga port on your motherboard ?) to be able to access the web at least.

Thanks everyone.

You're kidding? That is the average temp you know. That is not high by any means.

Well kidding about the replacing part, not the annoying fan part.
It's idling at anything from 65 to 83 when doing nothing at all.
But it's already five or so years old so I was planning on getting a new computer anyway, installing some dual boot linux and all that kind of stuff. Re-installing an old computer is annoying, installing a new one is exciting!

Oh you meant idle? Sorry I misread I guess. It's good, get a new one! ;)

Seems indeed you have cooling problems, follow instructions like people mentioned, clean card and fans with compressed air, check if fans are working, apply new thin layer or thermal paste, use a framelimiter or turn on v-sync.

There is another thing, you should check and see what the overclocksettings of your card are and lower them or set it to factory default if applicable. Yet another thing is that the GPU cooling fans might be worn and don't reach full RPM when needed, I'm not sure how to check that but try googleling for it. Last thing it might also be your PSU that isn't able to deliver enough power to the fans anymore, though not likely it is a possibility.

If you can afford it, then IMO a new card is the way to go. Even a Radeon R9 270X will be significantly faster than what you have, and even the more powerful AMD cards are currently not that expensive. If you have money for a high end card, then the GeForce 970/980 cards look like great solutions.

Kinda what I'm thinking. I would also remove the heatsink, apply new thermal paste, and reinstall. 5-year-old thermal paste is likely baked and useless by now.

hmm, ninja'd by a few hours. Oh well, add my voice to those suggesting replacement of the thermal compound.