  The bulk magnetic properties of the new quasi-one-dimensional Heisenberg
antiferromagnet, CuCrO$_4$, were characterized by magnetic susceptibility, heat
capacity, optical spectroscopy, EPR and dielectric capacitance measurements and
density functional evaluations of the intra- and interchain spin exchange
interactions. We found type-II multiferroicity below the N\'{e}el temperature
of 8.2(5) K, arising from competing antiferromagnetic nearest-neighbor ($J_{\rm
nn}$) and next-nearest-neighbor ($J_{\rm nnn}$) intra-chain spin exchange
interactions. Experimental and theoretical results indicate that the ratio
$J_{\rm nn}/J_{\rm nnn}$ is close to 2, putting CuCrO$_4$ in the vicinity of
the Majumdar-Ghosh point.
