Evidence is presented for the production of prompt $J\psi$ mesons (not originating in b-hadron decays) in hadronic Z⁰ decays. Using a sample of 3.6 million hadronic events, 24 prompt $J\psi$ candidates are identified from their decays into e⁺e⁻ and μ⁺μ⁻ pairs. The background is estimated to be 10.2 ± 2.0 events. The following branching ratio for prompt $J\psi$ production is obtained: $Br(Z$⁰ → promptJψ + X) = (1.9 +- 0.7 ± 0.5 ± 0.5) · 10⁻⁴, where the first error is statistical, the second systematic and the third error accounts for uncertainties in the prompt $J\psi$ production mechanism.