User Module modifications
Clive's answer is right for Drupal 6.

For Drupal 7, you'd implement a Field. This field would provide the 64-field form to fill out, the way to display the info, and know how to query the external database. (Obviously there's some room for design improvement here.. I'm chunking all the different pieces into one meta-field, but you could split it out however you want.)

Then you go to admin/config/people/accounts/fields and add your field to the user account profile. When you add it, there will be an initial configuration page for that field instance, and you'd check 'Display on user registration form.'

Easy as pie, right? :-)

There's a generic field implementation example in the Examples project.