The authors report on the fabrication and efficient continuous-wave operation, at 1.084 mum, of both x-cut and z-cut Nd:MgO:LiNbO3 channel waveguide lasers using a novel proton-exchange technique. The waveguide absorption near lambda = 600 nm is presented, in proton-exchange Nd:MgO:LiNbO3 waveguides, for the first time. Highly reflecting dielectric mirrors are directly coated onto the substrate end-faces, and the waveguide lasers are optically pumped with a dye laser at 592 nm, and a semiconductor-diode laser at 813 nm. Thresholds as low as 4 mW are achieved in the waveguide laser cavities. Up to 2.8 mW of output power (limited by the available pump power) can be achieved, without observation of photorefractive damage. For the semiconductor-diode laser pump, a system of beam-shaping optics is used to improve the coupling efficiency to the waveguide. A calculation of the absorbed pump power threshold, using known parameters, of 3.0 mW is in good agreement with the experimental value of 3.4 mW.