Why are you using absolute links? If you were using relative links, of the form /node/22, the problem would not arise.
I am assuming you are using Drupal 7. The answer for D6 would be a little more complex.
If the links are in the node body, and if you have direct access to your database, you can update the text by running a query on the
First, back up your database in case this goes wrong.
Then run the following query on your tables:
UPDATE field_data_body SET body_value = REPLACE( body_value, 'a href="http://www.mysite.com/','a href="http://mysite.net/');
Repeat with the
This should change all references to the old domain with the new domain.
If you have links in the comments, you might also want to do
field_revision_comment_body, scanning for references in the
In the event the links are being generated automatically by a text filter, change the REPLACE patterns accordingly.