We have performed smoothed particle magnetohydrodynamics (SPMHD) simulations demonstrating the production of collimated jets during collapse of 1 M molecular cloud cores to form the first hydrostatic core in low-mass star formation. Recently, a number of candidate first-core objects have been observed, including L1448 IRS2E, L1451-mm and Per-Bolo 58, although it is not yet clear that these are first hydrostatic cores. Recent observations of Per-Bolo 58 in particular appear to show collimated, bipolar outflows which are inconsistent with previous theoretical expectations. We show that low-mass first cores can indeed produce tightly collimated jets (opening angles <or equal to 10 degrees) with speeds of similar to 27 km s-1, consistent with some of the observed candidates. We have also demonstrated, for the first time, that such phenomena can be successfully captured in SPMHD simulations.