Do the new ATI HD6850 and HD6870 support double precision? Can I use those on this project?

No!

Not as far as we can determine. The first AMD HD69xx card supporting DP will be the HD6950 and then the HD6970 when these are released.

ATM the top of the pile for Milkyway crunching is the HD5870 and HD5850 (single cards). If you feel money flushed then the twin card HD5970/5990 will be the king, as TGG can testify.

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Go away, I was asleep