An electronic device enclosure is provided having a top and a bottom, preferably each made of a polycarbonate plastic material and having respective lower and upper edges which are the same shape, and an elastomeric cover which fits over the top and wraps around the lower edge thereof so as to seal the joint between the top and the bottom. Preferably, the cover is made of silicone rubber or urethane of a durometer in the range of 40 to 60. Push buttons with associated suspensions are moulded integrally into the cover so as to protrude downwardly through apertures in the top and engage push button switches on the electronic device therein. A display aperture may be provided in the top and, if so, a portion of the cover also wraps around the edges of the display aperture so as to be disposed between the display and the cover. A battery access door is provided which comprises a central, relatively ridged material surrounded by an elastomeric material which seals the joint between the battery access door and an aperture in the enclosure and which also provides a holder for the batteries.