This Letter presents a measurement of the inelastic proton-proton cross section using 60 mu b(-1) of pp collisions at a center-of-mass energy root s of 13 TeV with the ATLAS detector at the LHC. Inelastic interactions are selected using rings of plastic scintillators in the forward region (2.07 s > 10(-6), where M-X is the larger invariant mass of the two hadronic systems separated by the largest rapidity gap in the event. In this xi range the scintillators are highly efficient. For diffractive events this corresponds to cases where at least one proton dissociates to a system with M-X > 13 GeV. The measured cross section is compared with a range of theoretical predictions. When extrapolated to the full phase space, a cross section of 78.1 +/- 2.9 mb is measured, consistent with the inelastic cross section increasing with center-of-mass energy.