We have searched for new light gauge bosons produced in π0 and η decays by studying the kinematically well-constrained reactions and , where one π0 or the η decays through the emission of a single photon recoiling against a missing state X (where X is a long-lived weakly interacting particle or ). No signal has been observed and branching ratio upper limits of 6 × 10−5 at 90% C.L. have been obtained for masses of the gauge boson lying between ∼65 MeV and 125 MeV (π0 decay), and 6 × 10−5 at 90% C.L. for X masses between ∼200 MeV and 525 MeV (η decay). The π0-decay limit represents a factor of 4 to 8 improvement when compared to the existing limit, whereas the η decay limit has been measured for the first time, thereby extending the mx range from 130 MeV up to 525 MeV.