An improved apparatus for cutting out discs from a sheet of cotton or other flexible material employs a reciprocating punch assembly having a punch whose stem is held in a housing that aligns the punch with an aperture in a die. The punch carries a protruding spherical segment that presses the interposed sheet material against the rim of the aperture in the die. To adjust for misalignment of the punch with the die aperture, the punch is resiliently supported in the housing in a manner permitting the punch sufficient movement to enable its spherical segment to enter the die aperture and force the flexible material against the entire circumference of that aperture in one stroke of the punch assembly.