In this letter, we propose a new constant false alarm rate (CFAR) detector to accelerate the existing superpixel (SP)-based CFAR detectors for ship detection in synthetic aperture radar (SAR) images. In our method, we design a new density-censoring operation to rapidly identify background clutter SPs (BCSPs) with high densities before the local CFAR detection. In this way, a large number of non-informative BCSPs are removed without time-consuming calculation of decision thresholds, and only a few candidate ship target SPs (STSPs) are retained. This reduces the computational cost of the subsequent local CFAR detection and the number of false alarms produced by it. During the local CFAR detection process for the retained candidate STSPs, we also propose an improved method to define their neighboring clutter regions (for the calculation of decision thresholds) using BCSPs identified by the density-censoring operation. Experiments on measured SAR images validate that the proposed CFAR method reduces the computational cost of commonly used SP-based CFAR methods by 75%-96% with similar or better detection performance.