Unregister jQuery, register jQuery.min – something wrong?

So, I’m currently trying to replace the normal jquery.js with the minimized and better version jquery.min.js.

And to do that, the logical way would be to do:

wp_deregister_script( 'jquery' );
    wp_register_script( 'jquery', 'http://ajax.googleapis.com/ajax/libs/jquery/1.6/jquery.min.js');
    wp_enqueue_script( 'jquery' );

Seeing as this would do the follow:
1: unregister/deregister the built in jquery.
2: register the new jquery at that URL.
3: enqueue that new script with the jquery call.

But no… This doesn’t work no matter what I do. However, my next script to registe, jquery-ui.min.js works just fine whenever I just put in the following:

wp_register_script('jqui', 'http://ajax.googleapis.com/ajax/libs/jqueryui/1.5.3/jquery-ui.min.js');

And then do a:

<?php wp_enqueue_script( 'jqui' ); ?>

I’ve searched around, and I’ve found that I’m doing the right thing over and over again. But for the love of whatever’s holy out there, it will not work, not even when I try to make one depend on the other, or even if I promise my firstborn to the underworld.

If anyone could please shed some light on this issue, please, do so…