Azimuth ambiguities appear widely throughout spaceborne Synthetic Aperture Radar (SAR) images. If the ambiguous energy is relatively strong, a large number of brilliant areas or points will emerge, which may be erroneously judged as actual targets. This is a disadvantage in image interpretation. Due to the fact that ambiguous energy is mixed with energy from the main zone in the frequency and time domains, it is difficult to suppress azimuth ambiguity to a reasonable level using the existing approach without loss of resolution. This study proposes an innovative approach for suppressing azimuth ambiguity based on the compressive sensing recovery framework, in which the original image acts as prior information and the corresponding frequency spectrum truncated in a proper ratio acts as measurement information. With the proposed approach, highresolution low-ambiguity images can be obtained by iteration. We used simulation and satellite data to validate the effectiveness of this proposed approach in suppressing azimuth ambiguity.