Ultimate Weapons in Final Fantasy IX can be obtained in various ways.
After the fight with Tiamat in Memoria, search to the left before countinuing to the next screen. Near some rocks you should find Amarant's best weapons, the Rune Claws. They have 93 attack.
Go to Quan's Dwelling (West of Treno, a small cave in the mountains.) and keep on going through each of the screens. Eventually, you should reach a small balcony-like area. If your chocobo is upgraded enough (Mountain Red is the minimum level required), Zidane will notice bubbles in the water below (Press X at the edge). Use a Dead Pepper and Choco will jump in the water, giving you items including the Tiger Racket.
Do note that the Tiger Racket can be equipted by Garnet as well.
Freya's ultimate weapon is the Dragon's Hair, and is quite tricky to find. Go northwest of Oeilvert and search around for an invisible crack. Choco must be at least mountain red, and you must use a Dead Pepper in the correct spot for choco to dig and find it. Due to the fact it's invisible, you must be patient. Be stocked with 99 Dead Peppers before trying this.
Garnet's ultimate weapon is rather tricky to find as well. You must find the Ocean Chocograph, which is to the left of the map. It's left of a small more linear island, so use that as a benchmark. The hints should also be helpful (from the chocograph menu), so be sure to read them. Once you find the treasure, you will recieve miscellanious treasures including the Whale Whisker.
To get the Gastro Fork, Quina's ultimate weapon you must participate in frog catching at the marshes around the world until you catch 99. Once you have caught 99 Master will appear and challenge you to a fight. Beat him and you shall receive the Gastro Fork.
Possibly the hardest weapon to get in the game as well as the most powerful. To get Excalibur II you must get to Memoria on Disc 4 within 12 hours of playing the game. After beating Lich (under the 12 hour limit), one of the columns in the room will give you Exaclibur II. The Ragnarok is his second best weapon, and can be found in the Outer Island Chocograph. While on Choco, head to the island immediately to the right of the entrance to the Desert Palace. After some trial and error, you should eventually dig up the chocograph, and be rewarded with treasures, including the Ragnorak.
When you're in Memoria, just before you fight Lich look around on the platform before the top. Scale along the walls and press X, and eventually you'll find the Mace of Zeus, which teaches Vivi Doomsday. (It's the only weapon that does so, so be sure to get it.)
To get Zidane's ultimate weapon (conveniently named Ultima Weapon) you must be on Disc 4 and have a golden chocobo. Make sure you have a dead pepper and go back to the location where Shimmering Island used to be (you must be in the EXACT location). Use a Dead Pepper, and hopefully you'll hit the treasure, giving you a the Ultima Weapon.