I guess count(1) will be even faster:
$res = mysql_query("SELECT count(1) as `number` FROM `table1`");
$count = mysql_fetch_result($res,0,'number');
Although haven't tried the proposed methods, the first makes database fetch all the records and count them in the database, the second makes database fetch a separate field for all the records and count the number of results on the server.
As a rule of thumb the less data you fetch for a particular record the less time it will take therefore I'd vote for updated first method (fetching constant for every record and counting the number of constants fetched).