Two submillimeter/millimeter sources in the Barnard 1b (B1-b) core, B1-bN and B1-bS, have been studied in dust continuum, H13CO+ J = 1-0, CO J = 2-1, 13CO J = 2-1, and C18O J = 2-1. The spectral energy distributions of these sources from the mid-IR to 7 mm are characterized by very cold temperatures of Tdust< 20 K and low bolometric luminosities of 0.15-0.31 L☉. The internal luminosities of B1-bN and B1-bS are estimated to be <0.01-0.03 L☉ and ∼0.1-0.2 L☉, respectively. Millimeter interferometric observations have shown that these sources have already formed central compact objects of ∼100 AU sizes. Both B1-bN and B1-bS are driving the CO outflows with low characteristic velocities of ∼2-4 km/s. The fractional abundance of H13CO+ at the positions of B1-bN and B1-bS is lower than the canonical value by a factor of four to eight. This implies that a significant fraction of CO is depleted onto dust grains in the dense gas surrounding these sources. The observed physical and chemical properties suggest that B1-bN and B1-bS are in an earlier evolutionary stage than most of the known class 0 protostars. In particular, the properties of B1-bN agree with those of the first hydrostatic core predicted by the MHD simulations. The CO outflow was also detected in the mid-IR source located at ∼15'' from B1-bS. Since the dust continuum emission was not detected in this source, the circumstellar material surrounding this source is less than 0.01 M☉. It is likely that the envelope of this source was dissipated by the outflow from the protostar that is located to the southwest of B1-b.