An exotic annihilation of e(+)e(-) at rest into a single gamma and a weakly interacting neutral vector boson V-0 was searched for by measuring the related gamma energy under the condition that no additional gamma was detected by a hermetic detector. No evidence of such an annihilation was observed, resulting in the upper limits on the branching ratio of 4.2 x 10(-6) (m(V0) < 200 keV/c(2)) and 2.1 x 10(-5) (m(V0) < 2m(e) = 1022 keV/c(2)). These upper limits were converted to those on the branching ratio of an exotic three-body decay (gamma gamma V-0) of orthopositronium (o-Ps), i. e. 6.2 x 10(-6) (m(V0) < 200 keV/c(2)) and 1.7 x 10(-5) (m(V0) < 2m(e)). As a result, this mode was definitely excluded as the source of the reported discrepancy in the o-Ps decay rate.