I'm no expert but I think most, if not all, 802.11g routers are backward-compatible to 802.11b.
I'm using a Linksys (Cisco Systems) WCG200 that I've had for a couple years and it's been very reliable - much, much better than the old BEFW11S4 it replaced. I run both g and b wireless adapters through it but naturally the g gives better performance.
One thing you might look into is the possibility of hooking up a high-gain antenna to it. My router works OK in my home with the standard, dual antennas but, if it didn't, that's what I'd do. If you want to do a search, look for a LINKSYS HGA7T High Gain Antenna Kit. There may be others that would also work - not sure.
More fool stabilizer, please.