A sample of 3.73 million hadronic Z decays, recorded with the OPAL detector at LEP in the years 1991-95, has been used to search for a narrow resonance corresponding to the decay of the D*'+/-(2629) meson into D*+/- pi+ pi-. The D*+ mesons are reconstructed in the decay channel D*+-> D0 pi+ with D0 -> K- pi+. No evidence for a narrow D*'+/-(2629) resonance is found. A limit on the production of D*'+/-(2629) in hadronic Z decays is derived: f(Z -> D*'+/-(2629)) x Br(D*'+ -> D*+ pi+ pi-) < 3.1 x 10^-3 (95% C.L.)