AMD certainly hasn't made a big song and dance about it, but it clearly say on its website that it will support this old socket supporting DDR one with its new CPUs.
Upgrading from socket 939 to AM2 will cost you a lot of money as you need to buy a new motherboard, new CPU and new set of memory as you do need DDR 2 to make AM2 work. Now with 5000+ and FX 62 listed for 939 you will just be able to buy a new CPU and your old memory and board should support it.
You can check out what AMD says, here. You can clearly see that Athlon 64 X2 5000+ (socket AM2 and 939) is priced at while Athlon 64 FX-62 (socket AM2 and 939) will cost $1,031.
Here is a screenshot of the AMD website as well.

AMD is too important to afford itself a typo on its official site but we will try to confirm this officially. µ