The beam position monitor (BPM) system for PEP-II, the B-Factory under construction at SLAC, is described in this paper. The system must measure closed orbit for a 3-A multibunch beam and turn-by-turn position for a low-current single bunch injected in a 200-ns gap in the multibunch beam. A system that combines broadband and narrowband capabilities and provides data at high bandwidth was designed. It includes a filter-isolator box (FIB) that selects a harmonic of the bunch spacing (952 MHz) and absorbs the other frequency components; a CAMAC-based wideband I&Q demodulator, ADC, and signal processor that provides beam position information to the control system; and a calibrator that must work even in presence of beam, correcting for electronic measurement errors. This paper describes the system requirements, the electronics design, and the laboratory tests. Show Partial Abstract