As far as I can tell from the wikipedia article, the US doesn't have to /get rid/ of the nuclear weapons, it just can't "proliferate" them.
The International Court of Justice (ICJ), in its advisory opinion on the Legality of the Threat or Use of Nuclear Weapons, issued 8 July 1996, unanimously interprets the text of Article VI as implying that
"There exists an obligation to pursue in good faith and bring to a conclusion negotiations leading to nuclear disarmament in all its aspects under strict and effective international control."