While he does defeat her in a fierce battle in her underwater lair, he does not kill her directly. Instead, he mortally wounds her with his sword, Hrunting, but the final blow that kills her is delivered by a giant sword he finds within her lair.
Here's a breakdown of the events:
* Beowulf enters Grendel's mother's lair and fights her.
* He uses Hrunting, a sword loaned to him by Unferth, but it fails to penetrate her hide.
* Beowulf is forced to retreat and eventually finds a giant sword within her lair.
* He uses this sword to deliver the final blow that kills Grendel's mother.
This event highlights Beowulf's strength and courage, but also demonstrates the limits of human strength and the power of fate.