Hi,
Back in the old days of Mac OS 9, you could open an internet application and it would automatically fire up PPP and dial up the modem if I wasn't already connected (just using an iMac with a standard phone dial-up connection). Indeed, I think there is an option in Remote Access somewhere that says "allow applications to connect automatically when needed".
In Mac OS X 10.2.6, there is a similar option - but if selected, my computer just dials up continuously...when you boot it, as soon as you disconnect, etc etc. And with the option off, of course Safari, Mail etc just say "server cannot be found".
Network time is NOT turned on - is there anything else that could be hijacking the modem all the time?
Zen