We study a mechanism of generation of Majorana neutrino mass due to spontaneous breaking of chiral symmetry (SBCS) accompanied by the formation of a quark condensate. The effect of the condensate is transmitted to the neutrino sector via lepton-number violating (LNV) lepton-quark dimension-7 operators known in the literature as an origin of the neutrino-mass-independent mechanism of neutrinoless double-beta (0 nu beta beta) decay. The smallness of neutrino masses is due to a large ratio between the LNV scale and the scale of the SBCS. This is a new realization of the seesaw mechanism, which we dub the quark condensate seesaw (QCSS). We examine the predictions of the QCSS for 0 nu beta beta-decay and neutrino mass spectrum. We will show that our model predicts the normal neutrino mass ordering and narrow ranges of the neutrino masses.