Long-term (∼130 years) reconstruction of the interplanetary magnetic field (IMF) based on geomagnetic indices indicates that the solar wind magnetic field strength has a "floor," a baseline value in annual averages that it approaches at each 11 yr solar minimum. In the ecliptic plane at 1 AU, the IMF floor is ∼4.6 nT, a value substantiated by direct solar wind measurements and cosmogenic nuclei data. At high heliolatitudes, Ulysses measured a constant radial field with a magnitude (normalized to 1 AU) of ∼3.2 nT during solar minimum conditions in ∼1995 when the observed solar polar fields were ∼100 mT and in 2006 when the polar fields were ∼60 mT, as well as for solar maximum conditions in 2001 when the polar fields were close to zero. We identify the floor with a constant (over centuries) baseline open magnetic flux at 1 AU of ∼ Wb, corresponding 14 4 # 10 to a constant strength (∼10 11 A) of the heliospheric current. Solar cycle variations of the IMF strength ride on top of the floor. The floor has implications for (1) the solar wind during grand minima-we are given a glimpse of Maunder minimum conditions at every 11 yr minimum; (2) current models of the solar wind-both source surface and MHD models are based on the assumption, invalidated by Ulysses, that the largest scale fields determine the magnitude of the IMF; consequently, these models are unable to reproduce the high-latitude observations; and (3) the use of geomagnetic input data for precursor-type predictions of the coming sunspot maximum-this common practice is rendered doubtful by the observed disconnect between solar polar field strength and heliospheric field strength.