This is a bit complicated matter. The petitioner for this is a kid from a rich family, which is traveling by public airships just for fun. You can meet this family in every aerodrome. The boy bet with his mother that Deathgaze exist and asks you to prove it. Mother additionally asks you to take care of him.
This fight will take place aboard an airship, but before it does you'll have to visit almost every aerodrome in Ivalice and talk with this boy. To save up some money you can use Strahl. Every time you talk with him he'll tell you that the Deathgaze was even closer.
When you hear the boy saying something like this jump in any public ship and pick the course "Leisure Craft".
On the ship you'll hear a question for a volunteer to fight the Deathgaze. Agree and run to the upper deck.
about 190 000 HP
Resistant to Libra,
Vulnerable to Earth,
The enemy is surrounded with Paling from the beginning, so start the fight with some spells. Now the foe is using only Fireball and Crushing Fangs (a very strong attack which may cause death) and of course the normal attack, which also deals damage to the attacking monster. With time the Paling will fades and the enemy starts casting Bio, Curse, Reverse and Restore. Now you can attack him also with your weapons. At the end of the battle he'll also cast Magick Barrier on himself so you'll have to finish him off with your weapons.
Bounty: 3400 gil, 2x Elixir, Deathgaze's figure in Sky Pirate's Den.