In the Demon Ruins, past the second bonfire. Walk down the stairs leading to the Demon Firesage, but turn left instead of going forward at the intersection, and follow the path to the end, where the Large Flame Ember waits in a box. The way there is blocked by some Demonic Statues, as well as three Taurus Demons and some ambushing Burrowing Rockworms that surround the player immediately before reaching the chest. They will not attack the player as long as the player stays in the middle.
When given to Vamos, who is located near the end of The Catacombs, he gains the ability to further ascend fire weapons from +5 to +6, allowing for further fire weapon reinforcement from +6 to +10 (the highest).
Vamos claims that the Large Flame Ember is from New Londo, despite its location in the Demon Ruins.