The effective capacitance of a single Cooper-pair box can be defined as the second derivative of the energy with respect to gate voltage. This capacitance has two parts, the geometric capacitance, C_geom, and the quantum capacitance C_Q. The latter is due to the anti-level crossing caused by the Josephson coupling. It depends parametrically on the gate voltage and is dual to the Josephson inductance. Furthermore, it is negative in the ground state, with a magnitude that may be substantially larger than C_geom. A Cooper-pair transistor with an appropriate charging energy can behave like a Cooper-pair box when biased well below the superconducting energy gap. We have been able to detect C_Q in such a transistor, by measuring the in-phase and quadrature rf-signal reflected from a resonant circuit in which the transistor is embedded. C_Q can be used as the basis of a charge qubit readout by placing a Cooper-pair box in such a resonant circuit.