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The Wedding Ring is an item found in Paper Mario: The Thousand-Year Door.
It is necessary to find the Wedding Ring in order to progress the storyline. When in Keelhaul Key, Francesca will lose her wedding ring, and she will refuse to leave the island until it has been found. Mario and his partners must go out and search for the ring; it is eventually found near the location of the cave.
Later, Frankie posted a trouble at Trouble Center. He has lost the wedding ring. In the US version of Paper Mario: The Thousand-Year Door, it's lost in front of Garf's house in the easternmost area of Rogueport. In the Japanese version, it's lost on top of the wall guarded by Gus. If Mario takes the trouble, he'll receive the Gold Card.