Superconducting cavities with low RF frequencies and heavy damping of higher order modes (HOM) are desired for the main accelerator of High Energy Photon Source (HEPS), a 6 GeV synchrotron light source promising ultralow emittance currently under construction in Beijing. A compact 166.6 MHz superconducting cavity was proposed adopting a quarter-wave beta=1 geometry. Based on the successful development of a proof-of-principle cavity, a HOM-damped 166.6 MHz compact superconducting cavity was subsequently designed. A ferrite damper was installed on the beam pipe to reduce HOM impedance below the stringent threshold of coupled-bunch instabilities. Being compact, RF field heating on the cavity vacuum seal was carefully examined against quenching the NbTi flange. The cavity was later dressed with a helium vessel and the tuning mechanism was also realized. Excellent RF and mechanical properties were eventually achieved. Finally, the two-cavity string was designed to ensure smooth transitions among components and proper shielding of synchrotron light. This paper presents a complete design of a fully dressed HOM-damped low-frequency beta=1 superconducting cavity for HEPS.