  We describe the design, construction, and initial validation of the
variable-delay polarization modulator (VPM) designed for the PIPER cosmic
microwave background polarimeter. The VPM modulates between linear and circular
polarization by introducing a variable phase delay between orthogonal linear
polarizations. Each VPM has a diameter of 39 cm and is engineered to operate in
a cryogenic environment (1.5 K). We describe the mechanical design and
performance of the kinematic double-blade flexure and drive mechanism along
with the construction of the high precision wire grid polarizers.
