In the theory of star formation, the first hydrostatic core (FHSC) phase is a critical step in which a condensed object emerges from a prestellar core. This step lasts about one thousand years, a very short time compared with the lifetime of prestellar cores, and therefore is hard to detect unambiguously. We present IRAM Plateau de Bure observations of the Barnard 1b dense molecular core, combining detections of H 2 CO and CH 3 OH spectral lines and dust continuum at 2.3 " resolution (∼ 500 AU). The two compact cores B1b-N and B1b-S are detected in the dust continuum at 2mm, with fluxes that agree with their spectral energy distribution. Molecular outflows associated with both cores are detected. They are inclined relative to the direction of the magnetic field, in agreement with predictions of collapse in turbulent and magnetized gas with a ratio of mass to magnetic flux somewhat higher than the critical value, µ ∼ 2 − 7. The outflow associated with B1b-S presents sharp spatial structures, with ejection velocities of up to ∼ 7 km s −1 from the mean velocity. Its dynamical age is estimated to be ∼ 2000 yr. The B1b-N outflow is smaller and slower, with a short dynamical age of ∼ 1000 yr. The B1b-N outflow mass, mass-loss rate, and mechanical luminosity agree well with theoretical predictions of FHSC. These observations confirm the early evolutionary stage of B1b-N and the slightly more evolved stage of B1b-S.