A measurement of the inelastic proton-proton cross section with the CMS detector at a center-of-mass energy of $\sqrt{s} =$ 13 TeV is presented. The analysis is based on events with energy deposits in the forward calorimeters, which cover pseudorapidities of -6.6 $< \eta <$ -3.0 and +3.0 $< \eta <$ +5.2. An inelastic cross section of 68.6 $\pm$ 0.5 (syst) $\pm$ 1.6 (lumi) mb is obtained for events with $M_\mathrm{X} >$ 4.1 GeV and/or $M_\mathrm{Y} >$ 13 GeV, where $M_\mathrm{X}$ and $M_\mathrm{Y}$ are the masses of the diffractive dissociation systems at negative and positive pseudorapidities, respectively. The results are compared with those from other experiments as well as to predictions from high-energy hadron-hadron interaction models.