How to hide "Welcome: xxxx" phrase

A request, to make this very useful module more customizable, on how to avoid the display of "Welcome: ..." message.
Now there's no way to do it without modifying the module, because it is blocked in a non themable function, I can't neither use css to display none:

function ajax_get_login_block_content() {
  global $user;
  $user_uri = theme('my_account_link', array());
  $login_uri = theme('user_login_link', array(), array());
  $register_uri = theme('user_register_link', array(), array());
  $logout_uri = theme('user_logout_link', array());
  if (user_is_anonymous()){
    $block_content = "<div id='ajax_register_block_anonymous'>". $login_uri ."<br/>". $register_uri ."</div>";
    return $block_content;
  }
  else{
    $block_content = "<div id='ajax_register_block_authenticated'><p>".t("Welcome: ")."<span class='ajax_register_user_name'>". theme('username', $user) ."</span></p>". $user_uri ."<br/>". $logout_uri ."</div>";
    return $block_content;
  }
}