When he woke, he journeyed in the wilds until he came upon the Mirrormere, the lake that the Dwarves call Kheled-zâram. Looking upon mysterious stars reflected in its surface, he took this as a sign, and there founded a great city tunnelled in the heart of the Misty Mountains. This was Khazad-dûm, later called Moria.
Durin lived to a great age even by the measure of the Dwarves, but though he was called the Deathless, he was not immortal. He died some time before the end of the First Age, but he founded the line of the Kings of Durin's Folk that extended down through the history of Middle-earth.
Durin's name appears to come from the Old Norse for 'one who sleeps'. That would make sense, given that the Dwarves were set to sleep under the earth until after the first Elves appeared, and Durin was said to be the first of the Fathers of the Dwarves to awaken at that time.