A prototype multiwire proportional chamber (MWPC) was developed to demonstrate the feasibility of constructing a radiopure time projection chamber with MWPC track readout to assay materials for alpha- and beta-emitting surface contaminants for future rare-event-search experiments as well as other scientific fields. The design features and assembly techniques described here are motivated by the position and energy resolution required to reconstruct alpha and beta tracks while efficiently rejecting backgrounds. Results from a test setup using an $^{55}$Fe x-ray source indicate excellent operational stability and a near-ideal energy resolution of 15.8% FWHM at 5.89 keV and a gas gain of $sim$10$^{4}$.
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