I guess we should have defined "run" first. From my experiences with my old Athlon 64 with 1GB of RAM, Windows 7 ran fine as long as it was the 32-bit version. With the 64-bit version, the OS was much more sluggish and normal tasks took noticeably longer. I immediately reinstalled the 32-bit version and things ran much better. So it will "run" on either, but will the OS actually be usable on both with only 1GB of RAM, no, not in my opinion.
Asus Laptop: K53TA
OS: Windows 7 Home Premium, 64-Bit - SP1
CPU: AMD Llano A6-3400M, Quad-Core, 1.4GHz-2.6GHz (Overclocked)
GPU: AMD Radeon HD6650M, 1GB GDDR3 (Catalyst 13.1)
RAM: Samsung 4GB DDR3-1333
OS: Windows 7 Home Premium, 64-Bit - SP1
CPU: AMD Llano A6-3400M, Quad-Core, 1.4GHz-2.6GHz (Overclocked)
GPU: AMD Radeon HD6650M, 1GB GDDR3 (Catalyst 13.1)
RAM: Samsung 4GB DDR3-1333