A novel low-temperature bolometric method has been devised and implemented for high-precision measurements of the microwave surface resistance of small single-crystal platelet samples having very low absorption, as a continuous function of frequency. The key to the success of this nonresonant method is the in situ use of a normal metal reference sample that calibrates the absolute rf field strength. The sample temperature can be controlled independently of the 1.2 K liquid-helium bath, allowing for measurements of the temperature evolution of the absorption. However, the sensitivity of the instrument decreases at higher temperatures, placing a limit on the useful temperature range. Using this method, the minimum detectable power at 1.3 K is 1.5 pW, corresponding to a surface resistance sensitivity of ≈1 μΩ for a typical 1 mm×1 mm platelet sample.