The total photoabsorption cross section on a free proton was measured at the GRAAL facility in the energy range E-gamma = 600-1500 MeV. The large-aperture LAGRAN gamma E detector and a liquid hydrogen target were used in the experiment performed with a back-scattered Compton gamma beam. To improve the accuracy, two alternative methods were employed. First, a subtraction method of using empty-target measurements allowed the cross section sigma(tot) to be evaluated directly because of a low level of the electromagnetic background. Second, an algorithm for evaluating sigma(tot) on the basis of summing the dominating partial cross sections was developed. Experimental results obtained for sigma(tot) by the two methods are compared with existing data.