The 1959 Best Professional Magazine Hugo was presented September 7 at Detention. For the first time the modern nominations ballot followed by a final ballot process was followed so we have a list of nominees and a winner, but we do not have the order of finish after first place. (The category was called "Pro-Mag of 1958" on the final Hugo ballot.)
Winner: The Magazine of Fantasy & Science Fiction ed. by Anthony Boucher and Robert P. Mills
Astounding Science Fiction ed. by John W. Campbell, Jr.
New Worlds ed. by John Carnell
Galaxy ed. by H. L. Gold
Infinity Magazine ed. Larry Shaw