Molecular quantum gases (that is, ultracold and dense molecular gases) have many potential applications, including quantum control of chemical reactions, precision measurements, quantum simulation and quantum information processing(1-3). For molecules, to reach the quantum regime usually requires efficient cooling at high densities, which is frequently hindered by fast inelastic collisions that heat and deplete the population of molecules(4,5). Here we report the preparation of two-dimensional Bose-Einstein condensates (BECs) of spinning molecules by inducing pairing interactions in an atomic condensate near a g-wave Feshbach resonance(6). The trap geometry and the low temperature of the molecules help to reduce inelastic loss, ensuring thermal equilibrium. From the equation-of-state measurement, we determine the molecular scattering length to be + 220(+/- 30) Bohr radii (95% confidence interval). We also investigate the unpairing dynamics in the strong coupling regime and find that near the Feshbach resonance the dynamical timescale is consistent with the unitarity limit. Our work demonstrates the long-sought transition between atomic and molecular condensates, the bosonic analogue of the crossover from a BEC to a Bardeen-Cooper-Schrieffer (BCS) superfluid in a Fermi gas(7-9). In addition, our experiment may shed light on condensed pairs with orbital angular momentum, where a novel anisotropic superfluid with non-zero surface current is predicted(10,11), such as the A phase of He-3. A Bose-Einstein condensate of molecules is produced by pairing atoms in an atomic condensate; this transition is the bosonic analog of the Bardeen-Cooper-Schrieffer superfluid to BEC crossover in Fermi gases.