The moon is moving through space in free-fall and is therefore regarded as weightless. Weight, defined as the net force of gravity on an object, is not relative to an orbiting body such as the moon.
The moon does have mass, which can be taken as a stand-in for weight. The moon's mass is roughly 7.3477×10^22 kilograms, or 73.5 million million million metric tonnes. This is the figure that would most likely register on a standard scale if the moon could be brought to Earth and weighed and if the Earth's gravity acted equally on all parts of the moon.