In Morte d'Arthur, how does Arthur kill Mordred and what does Mordred do just before he dies?

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.