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We propose a modified two-tone method that could be used for sensitive measurements of the intrinsic microwave surface impedance (Z/sub S/) of thin superconductor films and the tan/spl delta/ of a low-loss dielectric. An open-gap resonator scheme is used to measure the penetration depth (/spl lambda/) of thin superconductor films and extract the intrinsic Z/sub S/ of the superconductor films from its measured R/sub S//sup eff/ and /spl lambda/. We use a very small gap of 10 /spl mu/m between the top plate and the rest parts of the resonator. The tan/spl delta/ of rutile in the low 10/sup -7/ range and the dielectric constant as high as /spl sim/110 are observed at temperatures below 10 K at /spl sim/15.2 GHz, which enable to measure the R/sub S/ of the 10 mm-in-diameter YBCO films as low as /spl sim/100 /spl mu//spl Omega/ at the same frequency (f). The discrepancy between the R/sub S//sup eff/ at /spl sim/15.2 GHz and that at /spl sim/8.5 GHz scaled to /spl sim/15.2 GHz appears less than 2% when the relations of R/sub S/ /spl prop/ f/sup 2/ and tan/spl delta/ /spl prop/ f are used. We describe usefulness of our measurement method for measuring the intrinsic microwave properties of various superconductor samples.