Arthur kills Mordred with a spear. The armies have been fighting all day until at, one point, Arthur sees Mordred and challenges him. He calls him a traitor and tells him that the day of his death has come. Then he stabs Mordred with the spear all the way through his body.
But just before Mordred dies, he inflicts a mortal wound on Arthur. He stabs Arthur in the head with his sword -- he does it so hard that the sword goes through Arthur's helmet and into his brain.
This all happens in Chapter IV of Volume II.
In Malory's, Morte d'Arthur, King Arthur runs Mordred through with his spear. He manages to get "under" Mordred's shield and drives the spear "throughout the body more than a fathom." A fathom is six feet.
Before he dies, however, Mordred, knowing he's going to die anyway,
",,,thrust himself with the might that he had up to the burr of king Arthur's spear, and right so he smote his father King Arthur with his sword holden in both his hands, upon the side of the head, that the sword pierced the helmet and the casing of the brain.
The burr is the hand guard. Thus, Arthur kills Mordred and Mordred kills Arthur.
Mordred is the main antagonist of Malory's Morte d'Arthur. He is the illegitimate son of Arthur and his half-sister Morgause, and deeply resents his illegitimate and incestuous birth. He is portrayed as motivated by bitterness and hatred. Merlin has prophesied that Mordred will eventually be Arthur's doom.
The final battle of the story culminates with an individual fight between Arthur and Mordred. In this battle, Arthur successfully manages to thrust a spear through Mordred's body, but as Mordred is dying from this wound, he manages to raise up his own sword and stab Arthur in the head, an injury that appears mortal, although Arthur survives long enough to command Sir Bedivere to throw Excalibur, his sword, into the lake. Arthur is then placed on a barge that is set on course for the magical land of Avilion. It is unclear whether that will be his tomb or whether he will be healed and return again in some mystical fashion.