  We are concerned with the one-dimensional piston problem for the compressible
Euler equations of Chaplygin gas. If the piston moves at constant subsonic
speed to the uniform gas, there exists an integral weak solution for the piston
problem, consisting of a shock separating constant states ahead of the piston.
While if the speed of the piston is sonic or supersonic, a singular measure
solution, with density containing a Dirac measure supported on the piston,
shall be introduced to solve the problem. Integral weak solution exists for the
piston receding from the gas with any constant speed, and there is no vacuum.
In the extreme case as the Mach number of the piston goes to infinity, the
limiting equations and solutions are the same as that for the polytropic gases.
