When you register the custom post type, you have to specify that the rewrite rule shouldn’t be prepended with the existing URL structure.
In short, this means that this line in your register_post_type call:
'rewrite' => array('slug' => 'projects'),
should turn into this:
'rewrite' => array('slug' => 'projects','with_front' => false),
For more info, check out the rewrite argument from the codex entry on register_post_type
edit: just make sure that, after updating the code, you flush the rewrite rules by visiting Settings > Permalinks. Otherwise you’ll still see the old links.