Dani,
My customer's MySql db's are tiny--nothing like your 1GB monster.

I simply use the
mysqldump command. As you most surely know, this command simply creates a text file that includes all the SQL statements to create your tables and indexes and insert all the data. For you, these files would be enormous. You would want to dump one table at a time. You'll probably want to use the --opt option which locks tables and disables buffering. (Otherwise, mysqldump tries to buffer the entire table to memory before writing to file--bad idea with your big tables.)
You can use the
mysqlimport program to import the mysqldump files.
I've not used
mysqlhotcopy, but it's supposed to be faster than mysqldump. It is only for MyISAM and ISAM tables--which you probably have. That's the default table type, and all my tables in my MySQL databases are MyISAM. I guess 5.0 supports the InnoDB type. With InnoDB databases, you can simply copy the raw files from one server to another (as long as both computers use the same floating point logic, etc.)
Hmmm...reading more about mysqlhotcopy....this may be a good solution for you. (At your linux server prompt, read the doc using `perldoc mysqlhotcopy`--worked for me anyway.) From the manpage:
DESCRIPTION
mysqlhotcopy is designed to make stable copies of live MySQL databases.
Here "live" means that the database server is running and the database may be in active use. And "stable" means that the copy will
not have any corruptions that could occur if the table files were simply copied without first being locked and flushed from within the server.
mysqlhotcopy supports scp copying--meaning you could copy directly to the new server, but this would keep the live server locked longer. Assuming you have enough disk space, the docs recommend copying to local disk, then scp to the new server "at your leisure". Then again, you really want the db locked until the new system is online. Like you said, you don't ever want any more updates to the old db once you start the copy process.
I just haven't had experience with any relatively large MySql databases. Please follow up and let us know your experience.