Two diamond-machining methods have been developed as part of an effort to design and fabricate an off-axis, biconic ellipsoidal, concave aluminum mirror for an infrared spectrometer at the Kitt Peak National Observatory. Beyond this initial application, the methods can be expected to enable satisfaction of requirements for future instrument mirrors having increasingly complex (including asymmetrical), precise shapes that, heretofore, could not readily be fabricated by diamond machining or, in some cases, could not be fabricated at all. In the initial application, the mirror is prescribed, in terms of Cartesian coordinates x and y, by aperture dimensions of 94 by 76 mm, placements of -2 mm off axis in x and 227 mm off axis in y, an x radius of curvature of 377 mm, a y radius of curvature of 407 mm, an x conic constant of 0.078, and a y conic constant of 0.127. The aspect ratio of the mirror blank is about 6. One common, "diamond machining" process uses single-point diamond turning (SPDT). However, it is impossible to generate the required off-axis, biconic ellipsoidal shape by conventional SPDT because (1) rotational symmetry is an essential element of conventional SPDT and (2) the present off-axis biconic mirror shape lacks rotational symmetry. Following conventional practice, it would be necessary to make this mirror from a glass blank by computer-controlled polishing, which costs more than diamond machining and yields a mirror that is more difficult to mount to a metal bench. One of the two present diamond machining methods involves the use of an SPDT machine equipped with a fast tool servo (FTS). The SPDT machine is programmed to follow the rotationally symmetric asphere that best fits the desired off-axis, biconic ellipsoidal surface. The FTS is actuated in synchronism with the rotation of the SPDT machine to generate the difference between the desired surface and the best-fit rotationally symmetric asphere. In order to minimize the required stroke of the FTS, the blanks were positioned at a large off-axis distance and angle, and the axis of the FTS was not parallel to the axis of the spindle of the SPDT machine. The spindle was rotated at a speed of 120 rpm, and the maximum FTS speed was 8.2 mm/s.