An all-quartz oscillating-disk viscometer of very high precision was used to measure the temperature dependence of the viscosities of methane and hydrogen sulfide at low densities. The measurements were based on a single calibration at room temperature with a value theoretically calculated using an accurate ab initio pair potential for helium and the kinetic theory of dilute monatomic gases. The uncertainty of the experimental data is conservatively estimated to be +/-0.15 % at room temperature increasing to +/-0.20% at the highest temperature of 682 K. The new data are compared with experimental data from the literature as well as with values obtained from the Quinones-Cisneros correlation for methane (2010) and the Schmidt correlation for hydrogen sulfide (2008) both implemented in the REFPROP computer program of the National Institute of Standards and Technology (NIST). The comparison shows that the low-density values of the Quinones-Cisneros correlation for methane, primarily based on measurements in the range from (211 to 392) K performed at NIST in 2007, can be considered as reference values. On the contrary, the new experimental data for hydrogen sulfide are (2 to 6) % lower than the low-density values of the Schmidt correlation and should be taken into account for an improvement of the viscosity correlation for hydrogen sulfide. Furthermore, the temperature dependence of the viscosities calculated theoretically using ab initio intermolecular potential energy hypersurfaces for methane and hydrogen sulfide and the extended kinetic theory of dilute molecular gases should be used to extrapolate the viscosity correlations to low and high temperatures.