There are four items you need to keep an eye out for: Doga’s Artiefact, Une’s Mirror, Griffin’s Heart, and the Rat Tail. Buy all four items, then find three nobles wandering around Treno who want to buy them from you. Two are near the Cafe Cardia (a duck-man in a cape, and a red mage), and a guy near the counter of the Snythesis Shop. Sell them their wanted items, then head back to the Auction again. (the duck-man you need to sell the first item, can’t remember which one, then speak with him again to sell him the second item.)
Lo and behold, you’ll find those four items once again for sale at the Auction, as well as a new item; the elusive Magical Fingertip. Buy that (you may as well get the other four items as two of them – Une’s Mirror and Doga’s Artiefact – will unlock a piece of music from FFIII at the Black Mage Village, but that’s another story), then head back to Dageurreo. Speak with the old man at the Inn again, and give him the Magical Fingertip. In gratitude, he’ll give you the Excalibur in return!
Incidently, you can keep a reign on the price an item is sold for, by doing a little trick. When an auction begins, head to the front of the audience near the Auctioneer, to where a ‘?’ icon appears over Zidane’s head, but don’t press anything just yet. Wait, and let the greedy Nobles bid back and forth, until it looks like an item is about to be sold. You can tell when this happens, when the Auctioneer says something like ‘2000 gil. Any more?’ When this happens, press X and bring up the bid screen, and only increase the current bid by 1 gil. Then press X again to confirm. This way, you’ll win the item for a lot less than if you’d actively participated in all the bidding previously.