Ultralow-loss concave micromirrors with radius of curvature below $60\mathrm{\mu m}$ were fabricated by laser ablation and reflective coatings. A 10-μm-long microcavity with a mode volume of $40{\mathrm{\mu m}}^3$ was set up with two such mirrors, and the cavity linewidth was measured both spectrally and temporally. The smallest linewidth obtained was $96\mathrm{MHz}$ , corresponding to a quality factor of $3.3\times {10}^6$ and a finesse in excess of $1.5\times {10}^5$ . With these parameters, we estimate that a variety of solid-state quantum emitters coupled to the cavity may enter the strong coupling regime.