Precessions of the angular distribution of numerous transitions in three normal deformation bands in194,193Hg, as well as of high-excitationM1/E2 transitions, were determined in a transient field experiment conducted at the Berkeley 88″ Cyclotron with the Gammasphere array. The reaction150Nd(48Ca,4n-5n)194,193Hg at a beam energy of 203 MeV was used to populate high-spin levels and to provide recoiling reaction-product nuclei with sufficient velocity to traverse a gadolinium ferromagnetic layer. The average precession of the observed γ-ray transitions reflects the precession of the magnetic moment of higher, unobservable quasicontinuum states which feed these states. The resulting averageg-factors obtained for the three main decay paths in each of the193,194Hg isotopes are: 〈g(193Hg)〉 = 0.19(1) and 〈g(194Hg)〉 = 0.26(1). For theM1/E2 transitions the results are: 〈g(193Hg)〉 = 0.24(5) and 〈g(194Hg)〉 = 0.26(4), respectively. The reduction of the averageg-factors in even isotopes compared to the naiveZ/A value, and the further reduction in odd isotopes, is evident. These results are discussed in terms of a neutron particle contribution and are compared to similar data in lighter Hf nuclei.