- We present Herschel observations of 62 Early-Type Galaxies (ETGs), including 39 galaxies morphologically classified as S0+S0a and 23 galaxies classified as ellipticals using SPIRE at 250, 350 and 500 microns (and PACS 100 and 160 microns for 19 sources) as part of the volume-limited Herschel Reference Survey. We detect dust emission in 24% of the ellipticals and 62% of the S0s. The mean temperature of the dust is 23.9+/-0.8 K, warmer than that found for late-type galaxies in the Virgo Cluster. Including the non-detections, the mean dust mass is log(Mdust) = 5.9+/-0.1 and 5.2+/-0.1 Msun for the S0s and elliptical galaxies respectively. The mean dust-to-stellar mass is log(Mdust/Mstar) = -4.4+/-0.1 (S0s) and -5.8+/-0.1 (ellipticals). Virtually all the galaxies lie close to the red sequence yet the large number of detections of cool dust, the gas-to-dust ratios and the ratios of far-infrared to radio emission all suggest that many ETGs contain a cool interstellar medium similar to that in late-type galaxies. The mean dust-to-stellar mass ratio for S0s is approximatly a factor of ten less than for early-type spirals and the sizes of the dust sources in the S0s are also much smaller. We show that the difference cannot be explained by either the different bulge-to-disk ratios or environmental effects such as ram-pressure stripping. The wide range in the dust-to-stellar mass ratio for ETGs and the lack of a correlation between dust mass and optical luminosity suggest that much of the dust in the ETGs detected by Herschel has been acquired as the result of gravitational interactions; these interactions are unlikely to have had a major effect on the stellar masses of the ETGs. The Herschel observations tentatively suggest that in the most massive ETGs, the mass of the interstellar medium is unconnected to the evolution of the stellar populations.