Problem importing SQl file to new database

Hi,
I’ve built a site on a test domain but now want to transfer it to the client’s server. I’ve exported the sql file from the test site database testsite_ and created a database newsite_ and user newsiteuser_ (with all privileges) on the new server. I’ve copied all the files across via ftp but when I come to import the old database I get the following error:

#1044 – Access denied for user ‘user’@’localhost’ to database ‘testsite_’

Does this mean I’m not importing the file as the right user? I can’t see a way to select users in phpMyAdmin.

I’m just starting out with wordpress and all this database stuff is completely new to me. I’m assuming I’ll have to change all instances of the old domain to the new one, but how and at what point do I do this? Any help would be massively appreciated.