Is it possible to notify users by e-mail when receiving a private message except when a specific user sends a message? I also opened a thread on this on Drupal.org but since it's been awhile I thought I'd try asking here.

Use Case: In general, I want users to receive notifications for new private messages.  I can do this using the included submodule for e-mailing private messages.

So that the private message inbox has a message when users log in for the first time, I used the Rules module to automatically send a "welcome" private message upon registration on behalf of an administrator account. However, this triggers the "send e-mail on private message" behavior.  Since users already are receiving a "please validate your e-mail address" e-mail, I don't want them to receive a second e-mail notifying them that they have received the "Welcome" private message.

Is there a way to prevent messages from certain users (or, alternately, certain messages) from triggering the e-mail notification?

I also looked into implementing this using rules instead of the included submodule but this would mean I couldn't use the functionality for reducing the number of notifications (currently, I have it set to not send additional e-mails if the user has yet to log in and read the first e-mail in a thread from a given user).
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;
    }
  }
}