A data sample corresponding to 1.23 million hadronic Z decays collected by the ALEPH detector at LEP has been searched for signals of the production of a non-minimal CP -even Higgs boson h in the reaction e + e − → hZ ∗ . The h decay modes considered were: those of the minimal standard model Higgs boson, with modified branching ratios; decays into a pair of CP -odd Higgs bosons A; and decays into invisible final states. Only one event was found, a very acoplanar e + e − pair which could originate from the standard model background process e + e − → e + e − v v ̄ . Upper limits for the cross-section of the reaction e + e − → hZ ∗ have been derived as a function of m h , the mass of the Higgs boson h. In the case of invisible decays, the 95% CL lower limit on m h is 65 GeV/ c 2 for a production cross-section equal to that of a minimal standard model Higgs boson. When combined with previous ALEPH results on the reaction e + e − → hA, these cross-section upper limits exclude a domain in the ( m h m A ) plane of the MSSM such that if invisible h and A decays can be neglected, 95% CL lower limits of 44 and 21 GeV/ c 2 result for m h and m A , respectively, independent of the other parameters of the model.