We use solar wind heat-flux electrons to determine the solar magnetic polarities of the interplanetary magnetic field (IMF) close to the heliospheric current sheet (HCS) around the time of the last solar minimum in 1995--1996. At that time the tilt angle of the HCS was very low and solar activity was minimal, allowing the Wind spacecraft to probe the polarities of the fields close to the HCS during a time relatively free of transient interplanetary coronal mass ejections (ICMEs). During the three periods we examined, all solar polarity boundaries predicted from Stanford source surface (SS) maps were observed. During 34 days in which the SS magnetic neutral line skimmed within 3¿ of the ecliptic, only six tangential excursions into opposite solar polarities were observed. The distribution of the durations of magnetic polarity sectors was very similar to that reported earlier for solar maximum in 1978--1982, showing no increase in sectors which might be expected for a spacecraft trajectory roughly tangential to a corrugated HCS separating regions of opposite polarity. The heat-flux electron pitch angle distributions, intervals of magnetic false polarities, high-latitude field excursions, and orientations of minimum variance vectors at polarity boundaries were all similar to those observed away from the boundaries. About one third of the polarity boundaries were displaced from the large-angle changes of the IMF direction usually thought to define the HCS. These observations suggest a single, globally smooth HCS in the form of a corrugated ribbon much less complex than and often separated from a thicker and more structured surrounding current sheet system formed by IMF discontinuities.