We demonstrate a superconducting (SC) microwave (mw) cavity that can accelerate the dark matter search by maintaining superconductivity in a high DC magnetic field. We used high-temperature superconductor (HTSC) yttrium barium copper oxide (YBCO) with a phase transition temperature of 90K to prevent SC failure by the magnetic field. Since the direct deposition of HTSC film on the metallic mw cavity is very difficult, we used the commercial HTSC tapes which are flexible metallic tapes coated with HTSC thin films. We fabricated resonating cavity ($f_{TM010}$ ~ 6.89 GHz) with a third of the inner wall covered by YBCO tapes and measured the quality factor (Q factor) at 4K temperature, varying the DC magnetic field from 0 to 8 tesla. There was no significant quality (Q) factor drop and the superconductivity was well maintained even in 8 tesla magnetic field. This implies the possibility of good performance of HTSC mw resonant cavity under a strong magnetic field for axion detection.