Magnetic fields are thought to play an important role in the formation of stars. However, that importance has been called into question by previous observations showing that B-fields in parsec-scale clouds and in 10,000 AU-scale dense cores are randomly aligned with respect to protostellar outflows, contradicting theoretical models of star formation that assume that bipolar outflows are parallel to the mean magnetic-field direction. To compare B-field and outflow directions at the 1000 AU protostellar envelope scale, we used the 1.3 mm full-Stokes polarimeter at CARMA to map dust polarization with ~2.5" resolution toward 16 low-mass protostars. The results show that magnetic fields on scales of ~1000 AU are not tightly aligned with outflows from the protostars. Rather, the data are consistent with scenarios where outflows and magnetic fields are preferentially misaligned (perpendicular), or where they are randomly aligned. If one assumes that outflows emerge along the rotation axes of circumstellar disks, then our results imply that the disks are not aligned with the fields in the cores from which they formed.