The standard proof of the Hahn-Banach theorem makes use of Zorn's lemma. I hear that, however, Hahn-Banach is strictly weaker than Choice. A quick search leads to many sources stating that Hahn-Banach can be proven using the ultrafilter theorem, but I cannot seem to find an actual proof. So...

1. What is the ultrafilter theorem; and
2. How does the said theorem imply the Hahn-Banach theorem?

Any easily-accessible reference would be quite enough; thanks in advance!