This Letter presents a measurement of the inelastic proton-proton cross section using 60 μb-1 of pp collisions at a center-of-mass energy 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<|η|<3.86) of the detector. A cross section of 68.1±1.4 mb is measured in the fiducial region ξ=MX2/s>10-6, where MX is the larger invariant mass of the two hadronic systems separated by the largest rapidity gap in the event. In this ξ range the scintillators are highly efficient. For diffractive events this corresponds to cases where at least one proton dissociates to a system with MX>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.