From Wikipedia:
Socrates’ death is described at the end of Plato’s Phaedo. Socrates turned down Crito’s pleas to attempt an escape from prison. After drinking the poison, he was instructed to walk around until his legs felt numb. After he lay down, the man who administered the poison pinched his foot; Socrates could no longer feel his legs. The numbness slowly crept up his body until it reached his heart.
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Actually, when I had to read about Socrates in college, I hated his personality but I could not figure out why until I found it in the Mesillat Yesharim (perek 13 IIRC). There are some people that are arrogant but feign humility, to not only be known as a great, but to be known as humble too. IMHO, that passage fits well with Socrates. He wanted to prove his cleverness and his humilty.