The branching fraction of the rare decay Lambda(0 )(b)-> Lambda(1520)mu(+)mu(-) is measured for the first time, in the squared dimuon mass intervals q(2), excluding the J/psi and psi(2S) regions. The data sample analyzed was collected by the LHCb experiment at center-of-mass energies of 7, 8, and 13 TeV, corresponding to a total integrated luminosity of 9 fb(-1). The result in the highest q(2) interval, q(2) > 15.0 GeV2/c(4), where theoretical predictions have the smallest model dependence, agrees with the predictions.