Sender address rejected

Send a test email to yourself.
This helps to clarify whether the message is coming from your outgoing mail server or the recipients incoming mail server.

Can you send a message to yourself from webmail?

Go to System Preferences>Accounts
Click on the lock and enter your password.
Click on the + sign above the lock.
Create a new user account.
Click on the apple menu and choose Log Out
Log into the newly created account.
Open Mail and enter your mail account settings.
Send a test email.
This will tell you if the issue is restricted to your user account or system wide.
If Mail works you know the issue lays within your user account.
If you still receive the message then the issue is either with the mail program, the existing system or a problem with the service provider.
If another computer from this same connection using the same mail settings can send mail then the issue is isolated to your computer.
If you isolate the issue as a system wide issue and have no other computer to test with let me know.
Sleep beckons.

You mention your previous computer. Is it a Mac and is it available?
You say the .Mac account is working, if you go to Mail>Preferences>Accounts>Server Settings, what port is being used?

If your mail was working or works on your iBook then you have a working plist.
When in the Finder >Go>Home>Library>Preferences
Hold down on the option key and drag the com.apple.mail.plist to your desktop, this creates a copy.
Mail the plist copy to yourself or otherwise move it to your new computer.
Navigate to Finder >Go>Home>Library>Preferences
Remove the existing com.apple.mail.plist from the folder and replace it with the one from your iBook. If Mail now works you will need to set up your .Mac account again. The port 587 is correct for .Mac. The fact that .Mac is working on port 25 means that BT are not blocking port 25 and this is not a port issue.
If the plist is not a possibility please test in a new user account as suggested earlier. If the issue is system wide you need to replace the Mail program and run the 10.4.10 Combo update or do an Archive and Install preserving User and Network settings. The Archive and Install is a better option if you do not have many 3rd party programs to reinstall. You would then need to run Software Update after a reinstall.

Try open Terminal in Utilities and cut and paste this command:
cat /var/log/mail.log
it will show you what is happening with your Mail. it might say 'unknown domain' or a message about authorization, etc. I had a problem with sbcglobal.net requiring me to use their smtp server. They would block port 25 unless I told them not to. You might have to change to port 587 for the SMTP outgoing in Mail Preferences. "Contact your mail hosting provider for information about using Port 587 for sending authenticated outgoing email. Since December 1998, Port 587 has been the Internet standard (RFC2476) port for authenticated email submission for mail clients and is supported by most major mail hosting providers. " good luck!