Custom permalink structure only for default posts

The solution is to reregister the default post type just after WordPress, and to add a rewrite slug. Also, the _builtin param needs to be set to false.

add_action( 'init', 'my_new_default_post_type', 1 );
function my_new_default_post_type() {

    register_post_type( 'post', array(
        'labels' => array(
            'name_admin_bar' => _x( 'Post', 'add new on admin bar' ),
        ),
        'public'  => true,
        '_builtin' => false, 
        '_edit_link' => 'post.php?post=%d', 
        'capability_type' => 'post',
        'map_meta_cap' => true,
        'hierarchical' => false,
        'rewrite' => array( 'slug' => 'post' ),
        'query_var' => false,
        'supports' => array( 'title', 'editor', 'author', 'thumbnail', 'excerpt', 'trackbacks', 'custom-fields', 'comments', 'revisions', 'post-formats' ),
    ) );
}