You'd need to modify your profile page code, to make it not show the editable areas, and not run the "update profile" action, if the user ID is [xyz].
For the page which actually does the updating of the profile, you can just put at the top something like
// Change this line to match however you identify your logged-in user
// And change the id number to the ID of the public user
global $current_user;
get_currentuserinfo();
if ($current_user->ID == 1)
{
// Stop them seeing this page
header('Location: index.php');
// And for good measure
die();
}
For the page on which they can change the profile fields before they submit the form, you can do something like this
// Change this line to match however you identify your logged-in user
// And change the id number to the ID of the public user
global $current_user;
get_currentuserinfo();
if ($current_user->ID == 1)
{
// Say no
echo '<p>You cannot edit your profile on this account.</p>';
// And for good measure
die();
}
Without seeing your code, it's hard to be more specific, but this should work at a push, even if it's not exactly how you want it to work.