A method is described to measure photoexcitation cross sections, relying on the expected behavior of the signal in the saturated regime, when excitation is provided by a Gaussian light beam. The method is implemented on a negative ion beam, with a single-mode pulsed Nd:YAG laser, to make a laser measurement of the photodetachment cross section of ${\mathrm{H}}^{\ensuremath{-}}$, at the wavelength 1064 nm. This cross section is of importance both as a photodetachment cross section of the most elementary negative ion and as a key parameter for the production of fast neutral ${\mathrm{H}}^{0}$ or ${\mathrm{D}}^{0}$ atoms, by photodetachment of accelerated anions. The obtained value $4.5(6)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}21}\phantom{\rule{0.16em}{0ex}}{\mathrm{m}}^{2}$ is greater than the one known from older measurements and most ab initio calculations.