Upgrade to 2.0.4 fails with fatal error in wp-db.php

I tried upgrading today from 1.5 following all the steps. Upon launching the upgrade.php script, I got:

Fatal error: mysql_num_fields(): supplied argument is not a valid MySQL result resource in wp-includes/wp-db.php on line 170.

This line reads:
while ($i < @mysql_num_fields($this->result)) {

Can anyone help, please?