Many modern theories predict that the fundamental constants depend on time, position or the local density of matter. Here we develop a spectroscopic method for pulsed beams of cold molecules, and use it to measure the frequencies of microwave transitions in CH with accuracy down to 3 Hz. By comparing these frequencies with those measured from sources of CH in the Milky Way, we test the hypothesis that fundamental constants may differ between the high- and low-density environments of the Earth and the interstellar medium. For the fine structure constant we find Δα/α=(0.3 ± 1.1) × 10⁻⁷, the strongest limit to date on such a variation of α. For the electron-to-proton mass ratio we find Δμ/μ=(-0.7 ± 2.2) × 10⁻⁷. We suggest how dedicated astrophysical measurements can improve these constraints further and can also constrain temporal variation of the constants.