We investigate the structure of the core surrounding the recently identified deeply embedded young stellar object Barnard 1c. B1c lies within the Perseus molecular cloud at a distance of 250 pc. It is a deeply embedded core of 2.4 M_⊙ (Kirk et al.) and a luminosity of 4 ± 2 L _⊙. Observations (and resolutions) of ¹²CO J - 1-0 (9″.2 × 5″9), ¹³CO J = 1-0, C¹⁸O J = 1 - 0 (14″.3 × 6″.7), HCO⁺ J = 1-0 (7″.6 × 5″.8), and N₂H⁺ J = 1-0 (5″.9 × 4″.6) were obtained with the BIMA array, together with the continuum at 3.3 mm (6″.4 × 4″. 9) and 2.7 mm (9″.5 × 6″.3). Single-dish measurements of N₂H⁺ J = 1-0 and HCO⁺ J = 1-0 with FCRAO reveal the larger scale emission in these lines with ∼ 60 resolution. The ¹²CO and HCO⁺ emission traces the outflow extending over the full field of view (2′.1), which coincides in detail with the S-shaped jet recently found in Spitzer IRAC imaging. The N₂H⁺ emission, which anti-correlates spatially with the C¹⁸O emission, originates from a rotating envelope with effective radius ∼2400 AU and mass 2.1-2.9 M_⊙, as derived from the 3.3 mm continuum emission. N₂H⁺ emission is absent from a 600 AU diameter region around the young star, offset from the continuum peak. The remaining N₂H⁺ emission may lie in a coherent torus of dense material. With its outflow and rotating envelope, B1c closely resembles the previously studied object L483 mm, and we conclude that it is a protostar in an early stage of evolution, i.e., Class 0 or in transition between Class 0 and Class I. We hypothesize that heating by the outflow and star has desorbed CO from grains, which has destroyed N₂H⁺ in the inner region, and surmise that the presence of grains without ice mantles in this warm inner region can explain the unusual polarization signature observed from B1c.