A gas cell apparatus to measure charge exchange cross sections for charge state- and energy-resolved ion beams with neutrals is described. The design features a short well-defined interaction region required for beams of multicharged ions with high cross sections. Our method includes measuring the beam transmission at four different neutral pressures and extracting the cross section from the slope of a beam loss vs pressure plot. The design and procedure were tested for Ar+ interacting with neutral Ar gas over the incident ion energy range of 1.0-5.0 keV. The charge exchange cross sections agree well with previous complementary measurement techniques.