A laser confocal radius measurement (LCRM) method is proposed for high-accuracy measurement of the radius of curvature (ROC). The LCRM uses the peak points of confocal response curves to identify the cat eye and confocal positions precisely. It then accurately measures the distance between these two positions to determine the ROC. The LCRM also uses conic fitting, which significantly enhances measurement accuracy by restraining the influences of environmental disturbance and system noise on the measurement results. The experimental results indicate that LCRM has a relative expanded uncertainty of less than 10 ppm for both convex and concave spheres. Thus, LCRM is a feasible method for ROC measurements with high accuracy and concise structures.