Is it possible to notify users by e-mail when receiving a private message except when a specific user sends a message?
Note: this is answer for Drupal 7.

As Letharion said, you can use hook_mail_alter() but you should detect somehow is this a welcome message from admin or not.
I suggest Context module for this task, but you really can just compare a subject of message in this hook.  

Update


  First I suggested to use Context module, but it is no so simple to set context in Rules. If you are interested in that way, you can see history of an answer.
  Current code is using information about Author id and subject of message:  


/**
 * Implements hook_mail_alter().
 */
function YOURMODULE_mail_alter(&amp;$message) {
  $WELCOME_USER_ID = 1; // PM author from your rule
  $WELCOME_SUBJECT = 'Welcome to the site'; // PM subject from your rule
  // 'pm_email_notify_notice' is an ID for message notifications
  // from Privatemsg Email Notification module
  if($message['id'] == 'pm_email_notify_notice') {
    $privatemsg = $message['params']['message'];
    if ($privatemsg-&gt;author-&gt;uid == $WELCOME_USER_ID &amp;&amp; $privatemsg-&gt;subject == $WELCOME_SUBJECT) {
    // If special context is set it means that we don't want to send an email
    //if(context_isset('context', 'my_privatemsg_stop_email')) {
      // Stop email
      $message['send'] = FALSE;
    }
  }
}