We report measurements of the inclusive production of heavy quarkonium states in Z decays based on the analysis of 3.6 million hadronic events collected by the L3 detector at LEP. The measurement of inclusive J production and an improved 95% confidence level upper limit on ϒ production are presented. In addition, two independent measurements of the ratio, f p , of prompt J mesons to those from B decay are made using two different isolation cuts to separate prompt J mesons from J mesons produced in the decays of b hadrons. The results are: Br ( Z → J + X )=(3.21±0.21 (stat.) +0.19 −0.28 (sys.) )×10 −3 , Br(Z→ ϒ (1S)+ X )<4.4×10 −5 , f p =(7.1±2.1 (stat.) ±1.2 (sys.) +1.5 −0.8 (theo.) )×10 −2 .