Yes, there's an open PCMCIA slot, and I may resort to it if I can find an inexpensive 56K dial-up modem to put into it. I'm frustrated, though, because both Device Manager and Belarc Advisor are telling me there's already a modem installed.
One weird feature of this laptop (I mean, besides the phantom modem): in the front is a sort of little door, with two flaps (upper and lower) that swing open in the same direction. That door is the one possible clue I can find to the modem's location. Each flap has two prongs on the inside, i.e., pointing into the laptop (four prongs in all). Maybe the four-pin dongle I've ordered will fit there and solve my problem. I'm just flailing, though. I really have no idea.
Anyway, thanks for your help so far. If you have other thoughts I'd appreciate them.