Contrary to popular belief, comets actually weigh nothing. Weight is a measure of how strongly gravity pulls on something, and since there is no gravity in space, the comet has no weight. A better measure would be the mass of a comet, which is independent of gravity.
The mass of a comet, which is a measure of how much matter is contained in it, varies from comet to comet. One of the few comets to be investigated in detail by scientists, 67P/Churyumov-Gerasimenko, was estimated to have a total mass of around 1x10^13 kg +/- 10 percent, or roughly 10 trillion kilograms. Other comets, however, are likely to have very different masses.