How is the weight of an airplane measured?
The empty weight of airplanes can only be given to single-pound precision for one specific airplane. Minute differences in metal sheet thickness, equipment and paint will make sure that there is a difference of many pounds between airplanes of the same type and furnishing. Add to this the difference in seating, galleys and instrumentation between airplanes of the same type which are operated by different airlines, and a precise number as in your question becomes almost meaningless. A better way would be to give a range in which most airplanes will fall, or to round the number like the Concorde weight of your question.

Weighing is quite straightforward: All tires of the landing gear are placed on scales and the weights are summed up. If the attitude of the airplane does not change, the weight on the tires can even be measured consecutively, so only one scale is needed. Small floatplanes can be suspended from a crane with a scale in the line holding the aircraft.

In Ernst Heinkel's biography he remembers an aircraft deal with the Soviet Union in the late 1920s. He had to deliver five floatplanes of a specified weight, and the Russian approval inspectors insisted on all planes having the exact same weight. Of course the planes were not identically heavy, but his advantage was that the scale in his factory was indicating the weight by printing it on cards. So the evening before the inspectors came, he let one aircraft be weighed five times, put the cards back into the machine and had the printing mechanism disabled. The next day he closed the business successfully with his very happy Russian guests.