like you i ues Azureus, have a 1mb connection, and have a thompson speedtouch 330. i had initial problems similar to you. what are the kinds of speeds you are getting, and what is your download, upload rate set at?
one more thing, do you download torrents with a high number of seeds?
well i would suggest bumping up your upload speed to about 35.
this is the speed that mine is set at and it ensures that you get a decent share ratio whilst not clogging up your line. if you still have no luck, post back. but im sure this is the reason. also some torrents take up to 10 minutes to pick up fast downloads.
Like Lethal said, u MUST pickup the upload speed and also Open Windows Firewall->Exceptions Tab->Add Port, and then add the listening port you are using in azareus, make sure it is NOT 6881-6889. This should stop the firewall from blocking you from connecting to a lot of seeds.