We present Cassini magnetic field observations from the only two close flybys (16DI and 129DI) of Saturn's icy satellite Dione which have been carried out so far. Data from 16DI show a weak field perturbation in the upstream region, indicative of a tenuous atmosphere around the satellite. By applying an analytical model of the perturbations caused by subalfvénic atmosphere-magnetosphere interactions, we demonstrate that an atmospheric column density of approximately 1 · 1017 m-2 would be able to sustain the observed field signature. Magnetic field data from 16DI also contain hints that Dione's gas envelope might possess a slight asymmetry between the Saturn-facing and the Saturn-averted hemisphere. The detection of a thin atmosphere at Dione might be correlated to the occurrence of a transient radiation belt near the moon's L-shell at the time of the 16DI flyby, as reported by Roussos et al. (2008b). On the other hand, magnetic field observations from the subsequent downstream encounter 129DI show no clear evidence of an atmosphere, probably due to the flyby trajectory being unsuitable for the detection of the associated perturbations.