CR-39 nuclear track material is frequently used for the detection of protons accelerated in laser-plasma interactions. The measurement of track densities allows for determination of particle angular distributions, and information on the kinetic energy can be obtained by the use of passive absorbers. We present a precise method of measuring spectral distributions of laser-accelerated protons in a single etching and analysis process. We make use of a one-to-one relation between proton energy and track size and present a precise calibration based on monoenergetic particle beams. While this relation is limited to proton energies below 1 MeV, we show that the range of spectral measurements can be significantly extended by simultaneous use of absorbers of suitable thicknesses. Examples from laser-plasma interactions are presented, and quantitative results on proton energies and particle numbers are compared to those obtained from a time-of-flight detector. The spectrum end points of continuous energy distributions have been determined with both detector types and coincide within 50-100 keV.