Events where the two leading jets are separated by a pseudorapidity interval devoid of particle activity, known as jet-gap-jet events, are studied in proton-proton collisions at √s = 13 TeV. The signature is expected from hard color-singlet exchange. Each of the highest transverse momentum (p_T) jets must have p_T^jet > 40 GeV and pseudorapidity 1.4 < |η^jet| < 4.7, with η^jet1η^jet2 < 0, where jet1 and jet2 are the leading and subleading jets in p_T, respectively. The analysis is based on data collected by the CMS and TOTEM experiments during a low luminosity, high-β^* run at the CERN LHC in 2015, with an integrated luminosity of 0.66 pb^-1. Events with a low number of charged particles with p_T > 0.2 GeV in the interval |η| < 1 between the jets are observed in excess of calculations that assume only color-exchange. The fraction of events produced via color-singlet exchange, f_CSE, is measured as a function of p_T^jet2, the pseudorapidity difference between the two leading jets, and the azimuthal angular separation between the two leading jets. The fraction f_CSE has values of 0.4-1.0%. The results are compared with previous measurements and with predictions from perturbative quantum chromodynamics. In addition, the first study of jet-gap-jet events detected in association with an intact proton using a subsample of events with an integrated luminosity of 0.40 pb^-1 is presented. The intact protons are detected with the Roman pot detectors of the TOTEM experiment. The f_CSE in this sample is 2.91 ± 0.70(stat)_-1.01^+1.08(syst) times larger than that for inclusive dijet production in dijets with similar kinematics.