The bulk magnetic properties of the new quasi-one-dimensional Heisenberg antiferromagnet, CuCrO4, were characterized by magnetic susceptibility, heat capacity, optical spectroscopy, EPR and dielectric capacitance measurements and density functional evaluations of the intra-and inter-chain spin exchange interactions. We found type-II multiferroicity below the Néel temperature of 8.2(5) K, arising from competing antiferromagnetic nearest-neighbor (Jnn) and next-nearest-neighbor (Jnnn) intra-chain spin exchange interactions. Experimental and theoretical results indicate that the ratio Jnn/Jnnn is close to 2, putting CuCrO4 in the vicinity of the Majumdar-Ghosh point.