We investigate the hard-square lattice-gas model by means of transfer-matrix calculations and a finite-sizescalinganalysis. Using a minimal set of assumptions we find that the spectrum of correction-to-scaling exponentsis consistent with that of the exactly solved Ising model, and that the critical exponents and correlationlengthamplitudes closely follow the relation predicted by conformal invariance. Assuming that these spectraare exactly identical, and conformal invariance, we determine the critical point, the conformal anomaly, and thetemperature and magnetic exponents with numerical margins of 10211 or less. These results are in a perfectagreement with the exactly known Ising universal parameters in two dimensions. In order to obtain this degreeof precision, we included system sizes as large as feasible, and used extended-precision floating-point arithmetic.The latter resource provided a substantial improvement of the analysis, despite the fact that it restrictedthe transfer-matrix calculations to finite sizes of at most 34 lattice units.