A clamping device and a machining method. The clamping device comprises: a base; a clamping head comprising a blind rivet and a locking member, the blind rivet comprising a middle column segment and enlarged portions at two ends, and the locking member being connected to an upper end of the blind rivet and defining a clamping opening; a lower cavity connected to the base; an upper cavity comprising an annular peripheral wall and a top wall having a central hole, the top wall being clamped between the two enlarged portions, the middle column segment of the blind rivet movably passing through the central hole, and the upper cavity being connected to the lower cavity in a fit manner; an elastic member located in the peripheral wall and supported between the enlarged portion at a lower portion of the blind rivet and the top wall; and a support cylinder movably fitted in the lower cavity, the support cylinder being supported below the elastic member.