We report the first largely model independent measurement of charged particle multiplicities in quark and gluon jets, Nq and Ng, produced at the Fermilab Tevatron in pp¯ collisions with a center-of-mass energy of 1.8 TeV and recorded by the Collider Detector at Fermilab. The measurements are made for jets with average energies of 41 and 53 GeV by counting charged particle tracks in cones with opening angles of θc=0.28, 0.36, and 0.47 rad around the jet axis. The corresponding jet hardness Q=Ejetθc varies in the range from 12 to 25 GeV. At Q=19.2  GeV, the ratio of multiplicities r=Ng/Nq is found to be 1.64±0.17, where statistical and systematic uncertainties are added in quadrature. The results are in agreement with resummed perturbative QCD calculations.