If you're running at least 4.3, you can set a Bcc additional header. PHP will interpret this internally before sending the message.
$headers = "Bcc: address@place.com\r\nFrom: me@place.com"; // etc
mail( $to, $subject, $message, $headers );
See the docs: http://www.php.net/manual/en/function.mail.php
Also note that because PHP doesn't expand escape sequences (\r, \n) when using single quotes, you must use double quotes for this string.