A method for computing electromagnetic properties of hadrons in quenched lattice QCD is applied to the extraction of electromagnetic properties of the octet baryons. This allows a determination of the full dependence of the baryon masses on the charges and masses of the valence quarks. Results of a first numerical study (at β = 5.7 with Wilson action and light quark masses fixed from the pseudoscalar meson spectrum) are reported. The quenched octet baryon isomultiplet splittings (with statistical errors) are found to be: N - P = 2.08 (+/-0.56), Σ0 - Σ+ = 2.63(+/-0.39), Σ- - Σ0 = 4.90(+/-0.36) and Ξ- - Ξ0 = 5.58 (+/-0.24) MeV. Estimates of the systematic electromagnetic corrections arising from finite volume are included in these results.