In recent years, the development of new lithium niobate electro-optic modulator designs and material processing techniques have contributed to support the increasing need for faster optical networks by considerably extending the operational bandwidth of modulators. In an effort to provide higher bandwidths for future generations of networks, we have developed a lithium niobate electro-optic phase modulator based on a coplanar waveguide ridged structure that operates up to 300 GHz. By thinning the lithium niobate substrate down to less than 39 µm, we are able to eliminate substrate modes and observe optical sidebands over the full millimeter-wave spectrum.