This article is from the Final-Fantasy and other Square Soft Games FAQ, by firstname.lastname@example.org (Nick Zitzmann) with numerous contributions by others.
There are two ways...
When the battle begins, cast Barrier and Haste on yourself (or use Big
Guard, if you've got it). Have your characters make physical attacks,
and don't forget to summon Bahamut now that you have that materia.
Although Demon's Gate can cause a lot of damage, this will help to raise
your Limit bars. To further this effect, make sure that you have used
Hypers on your characters prior to the battle. Use Aeris' 'Healing Wind'
break when you need it, and the other characters should always attack
with their best Break when the situation arises. I suggest that you have
a Restore materia attached to an All materia so that you can heal your
characters as needed. The Demon's Gate is level 45, and has 10000 HP and
400 MP. If you need more MP, use Ghost Hands or the Magic Hammer skill
to siphon MP from the boss.
Are you a pacifist? Try using Phoenix Downs on it to reduce it to one
HP/kill it outright, depending on the situation.