The Atom SoCs are soldered to the motherboard, so you should try to look for a specific motherboard with such a CPU on it. A quick search on Supermicro's website turned up the X9SBAA-F, which is a mini-ITX Atom S1260 based unit. Several vendors offer them for sale, such as
Ewiz/SuperBiiz. I have bought from that vendor before and they simply get Supermicro to drop ship stuff right to your door in non-bulk quantities. That's about as good of a setup as you can hope for in trying to buy uncommon server stuff in individual quantities and not pay out the nose for it. I ordered an unusual quad socket Opteron Supermicro whitebox board from them and it arrived in a handful of days for about a hundred bucks less than the handful of other websites that stock such unusual parts in lots of less than 10.