The Tech Report writes this new model will be just 0.63" - 0.83" thick while the Eee PC 1000 measures about 1.5" at its thickest point. The S101 model will feature a 10.2" display, Intel Atom 1.6GHz processor, Intel 945 GME chipset and 32GB or 40GB SSDs. Unfortunately they won't be cheap, the source claims these models will cost between $699 and $899.
The good news is that Asus has a thinner Eee PC in store. HKEPC has posted some details and a couple of photos of the Eee PC S101, which looks like a fancier cousin of the 1000. The S101 will have a 10.2" display, a 1.6GHz Intel Atom processor, an Intel 945 GME chipset, and either 32GB or 40GB solid-state drives depending on the exact variant. Those specs look similar to those of the Eee PC 1000, but HKEPC claims the S101 will be just "16 ~ 21mm" thick (about 0.63"-0.83").
What's the bad news? HKEPC goes on to say 32GB and 40GB variants of the Eee PC S101 will cost a respective $699 and $899 when they come out in mid-September.