Ion-beam figuring (IBF) is a precise surface finishing technique used for the production of ultra-precision optical surfaces. In this study, we propose an effective one-dimensional IBF (1D-IBF) method approaching sub-nanometer root mean square (RMS) convergence for flat and spherical mirrors. Our process contains three key aspects. First, to minimize the misalignment of the coordinate systems between the metrology and the IBF hardware, a mirror holder is used to integrate both the sample mirror and the beam removal function (BRF) mirror. In this way, the coordinate relationship can be calculated using the measured BRF center. Second, we propose a novel constrained linear least-squares (CLLS) dwell time calculation algorithm combined with a coarse-to-fine scheme to ensure that the resultant nonnegative dwell time closely and smoothly duplicates the required removal amount. Third, considering the possible errors induced by the translation stage, we propose a dwell time slicing strategy to divide the dwell time into smaller time slices. Experiments using our approaches are performed on flat and spherical mirrors as demonstrations. Measurement results from the nano-accuracy surface profiler (NSP) show that the residual profile errors are reduced to sub-nanometer RMS for both types of mirrors while the surface roughness is not affected by the figuring process, demonstrating the effectiveness of the proposed 1D-IBF method for 1D high-precision optics fabrication.