© 2021 CERN, for the CMS and TOTEM Collaboration.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 (pT) jets must have pTjet>40 GeV and pseudorapidity 1.4<|ηjet|<4.7, with ηjet1ηjet2<0, where jet1 and jet2 are the leading and subleading jets in pT, 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 pT>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, fCSE, is measured as a function of pTjet2, the pseudorapidity difference between the two leading jets, and the azimuthal angular separation between the two leading jets. The fraction fCSE 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 fCSE 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.