Are All The Ants As Heavy As All The Humans?

It is often said that if we were to weigh all the ants in the world, they would weigh as much as all of the people. This claim was originally made by Harvard University professor Edward Wilson and the German biologist Bert Hoelldobler, in their 1994 book Journey To The Ants.
They based their estimate on an earlier one by British entomologist C. B. Williams, who once calculated that the number of insects alive on earth at a given moment was one million trillion. But is this statement true?