The strongest transitions of Zn and Cr II are the most sensitive to relative variations in the fine-structure constant (Δα/α) among the transitions commonly observed in quasar absorption spectra. They also lie within just 40 Å of each other (rest frame), so they are resistant to the main systematic error affecting most previous measurements of Δα/α long-range distortions of the wavelength calibration. While Zn and Cr II absorption is normally very weak in quasar spectra, we obtained high signal-to-noise, high-resolution echelle spectra from the Keck and Very Large Telescopes of nine rare systems where it is strong enough to constrain Δα/α from these species alone. These provide 12 independent measurements (three quasars were observed with both telescopes) at redshifts 1.0-2.4, 11 of which pass stringent reliability criteria. These 11 are all consistent with Δα/α = 0 within their individual uncertainties of 3.5-13 parts per million (ppm), with a weighted mean Δα/α= 0.4 ± 1.4stat ± 0.9sys ppm (1σ statistical and systematic uncertainties), indicating no significant cosmological variations in α. This is the first statistical sample of absorbers that is resistant to long-range calibration distortions (at the <1 ppm level), with a precision comparable to previous large samples of ∼150 (distortion-affected) absorbers. Our systematic error budget is instead dominated by much shorter range distortions repeated across echelle orders of individual spectra.