Over the past decades, there have been extensive research endeavors in spectrum auction design. However, most solutions only focus on the allocation efficiency while ignoring the privacy leakage inherent in the process of spectrum auction. So far, the very few existing works on secure spectrum auctions either provide inadequate privacy protection or incur performance loss in terms of spectrum reusability. In this paper, for the first time, we propose PS-TAHES, a privacy-preserving and truthful double auction mechanism for heterogeneous spectrum. PS-TAHES is constructed based on our carefully designed security primitives, which can support various arithmetics over encrypted data, including multiplication, bid comparison, and sorting matrix, and they are well applicable in other contexts. We theoretically analyze the security and efficiency of PS-TAHES, which is proved to ensure a full and strong privacy protection for bidders while preserving the allocation efficiency of the original auction mechanism. Experimental results, consistent with the theoretical analysis, further validate the practical use of PS-TAHES in real-world applications.