The Black Cauldron

Taran (Grant Bardsley) is assistant pig-keeper on the small farm of Caer Dallben, home of Dallben the enchanter (Freddie Jones). Taran dreams of becoming a great warrior, but must stop daydreaming because his charge, the oracular pig Hen Wen, is in danger. The Horned King (John Hurt), a fearsome, skeletal, undead king who wears antler horns on his head, hopes she will help him find the Black Cauldron, which has the power to restore a kind of life to the dead, as undead slaves called "the Cauldron-Born", which he will use to rule the world. Dallben directs Taran to take Hen Wen to safety, but the lad's negligence results in the pig's capture by the Horned King's forces.
Taran follows them to the Horned King's stronghold and acquires the small, pestering companion Gurgi (John Byner) along the way. Taran leaves Gurgi to sneak into the castle and rescues Hen Wen, who flees, but he is captured himself and thrown into the dungeon, soon to be released by Princess Eilonwy (Susan Sheridan), a girl his age who is also trying to escape. In the catacombs beneath the castle, Taran and Eilonwy discover the ancient burial chamber of a king, where he arms himself with the king's sword. It contains magic that allows him effectively to fight the Horned King's minions and so to fulfill his dream of heroism. Along with a third captive, the comical, middle-aged bard Fflewddur Fflam (Nigel Hawthorne), they escape the castle and are soon reunited with Gurgi.
Following Hen Wen's trail, the four stumble into the underground kingdom of the Fair Folk, small fairy-like beings who reveal that Hen Wen is under their protection. When the cheerful, elderly King Eiddileg (Arthur Malet) reveals that he knows where the cauldron is, Taran resolves to go destroy it himself. Eilonwy, Fflewddur, and Gurgi agree to join him and Eiddileg's obnoxious right-hand man Doli (John Byner) is assigned to lead them to the Marshes of Morva while the Fair Folk agree to escort Hen Wen safely back to Caer Dallben. At the marshes they learn that the cauldron is held by three witches, the grasping Orddu (Eda Reiss Merin), who acts as leader, the greedy Orgoch (Billie Hayes), and the more benevolent Orwen (Adele Malis-Morey), who falls in love with Fflewddur at first sight; they cause a frighted Doli to abandon the group. Orddu agrees to trade the cauldron for Taran's sword, and he agrees, although he knows that to yield it will cost his chance for heroism. Before vanishing, the witches reveal that the cauldron is indestructible, and that its power can be broken only by someone who climbs in under his own free will, which will kill him. None of the companions will do that, so it seems Taran has traded his sword for nothing. Taran feels foolish for aspiring to destroy the cauldron alone, but his longer companions show their belief in him, and it appears that he and Eilonwy will kiss.
The Horned King's soldiers interrupt, finally reaching the marshes themselves. They seize the cauldron and everyone but Gurgi, and return to the castle. The Horned King uses the cauldron to raise the dead and his Cauldron-Born army begins to pour out into the world.
Gurgi manages to free the captives and Taran resolves to cast himself into the cauldron, but Gurgi stops that and advances himself instead. The undead army collapses. When the Horned King spots Taran at large, he infers the turn of events and throws the youth toward the cauldron, but the cauldron's magic is out of control. It consumes the Horned King and destroys the castle, using up all its powers.
The three witches come to recover the now inert Black Cauldron. Taran has finally realized Gurgi's true friendship, however, and he persuades them to revive the wild thing in exchange for the cauldron, giving up his magical sword permanently. Fflewddur goads the reluctant witches to go ahead and demonstrate their powers by the revival, which they do.
The four friends journey back to Caer Dallben where Dallben and Doli watch them in a vision created by Hen Wen, and Dallben finally praises Taran for heroism.

Question: Where do Taran and Eilonwy discover the ancient burial chamber of a king?
Answer: In the catacombs