A classic electrostatic induction machine, Cavallo's multiplier, is suggested for in situ production of very high voltage in cryogenic environments. The device is suitable for generating a large electrostatic field under conditions of very small load current. Operation of the Cavallo multiplier is analyzed, with quantitative description in terms of mutual capacitances between electrodes in the system. A demonstration apparatus was constructed, and measured voltages are compared to predictions based on measured capacitances in the system. The simplicity of the Cavallo multiplier makes it amenable to electrostatic analysis using finite element software, and electrode shapes can be optimized to take advantage of a high dielectric strength medium such as liquid helium. A design study is presented for a Cavallo multiplier in a large-scale, cryogenic experiment to measure the neutron electric dipole moment.