Five characteristics determine the utility of an x-ray optics technology for astronomy: (1) angular resolution, (2) field of view, (3) energy bandwidth, (4) mass per unit photon collecting area, and (5) production cost per unit photon collecting area. These five desired characteristics are always in conflict with each other. As a result, every past, current, and future x-ray telescope represents an astronomically useful compromise of these five characteristics. In this paper, we outline and report the proof of concept of a new approach of using single-crystal silicon to make lightweight x-ray optics. This approach combines the grinding polishing process, which is capable of making diffraction-limited optics of any kind, with the stress-free nature of single-crystal silicon, which enables post-fabrication light-weighting without distortion. As such this technology has the potential of making diffraction-limited lightweight x-ray optics for future astronomical missions, achieving unprecedented performance without incurring prohibitive mass and cost increase.