A spindle for machining at high rotational speeds, includes a tool-holding taper collet with radial slits that is kept closed to clamp a tool by pressing it axially against a seat of complementary shape by the reaction force of precompressed elastic elements interposed between the tool-holding collet and the body of the spindle, and is opened by moving it axially away from the seat by applying an axial force to it to further compress the precompressed elastic elements. In the spindle the elastic means are a sleeve of elastic material housed coaxially between the inside wall of a cavity formed in the body of the spindle and a shaped collar of epoxy resin applied coaxially around the outer surface of the free end of the tool-holding collet.