The monoblock surface milling cutter has a milling cutter body (10) to which cutting plates (16) made of polycrystalline diamond (PCD), monocrystalline diamond (MCD) or polycrystalline boron nitrite (PCB) are fastened in a material-to-material fashion. In order to improve the surface quality of the work pieces manufactured with the milling cutter, axially adjustable finishing inserts (18) are fixed in place in at least two substantially axially extending bores (26), which are open at the front, of the milling cutter body (10). The finishing inserts (18) can also be equipped with cutting plates made of PCD, MCD or PCB. Preferably the cutting edges (24) of the cutting plates (22) of the finishing insert (18) are forward curved. Their farthest forward point is located in relation to the axis of rotation of the milling cutter body (10) on a finishing circle diameter which is maximally 4 to 8 mm less than the diameter created by the cutting edges arranged on the exterior circumference of the milling cutter body (10).