I've called my ISP and double-checked all of my settings with them. They suggested that it was a "routing problem" through the telephone lines and that I should call my phone company. I did that: they said that since I could connect properly with my desktop computer on the same phone line, and since I heard the modem sound when I used my regular telephone to call the ISP's dial-up number, the problem must be in my laptop computer somewhere. I bought an external modem for the laptop - thinking that the problem was in the laptop's internal modem. (I installed the new external modem with its own software [on CD] and ran its special diagnostics -- all checked out OK.) -- This morning I double-checked all of the ISP dial-up connection settings on both computers; they are identical. I can connect perfectly on the desktop (the one that I'm using right now) -- but, when I use the same phone line and module as the desktop, the laptop still gives me "If you want to make a call, hang up and dial again..."
Since neither the technical people at my ISP nor at the telephone company seem to be able to help me, I thought I'd give Computer Hope a try. [Still hoping!] Thanks for any help you can give me.
