The cross section of the process e+e-→K+K- is measured at a number of center-of-mass energies s from 2.00 to 3.08 GeV with the BESIII detector at the Beijing Electron Positron Collider (BEPCII). The results provide the best precision achieved so far. A resonant structure around 2.2 GeV is observed in the cross section line shape. A Breit-Wigner fit yields a mass of M=2239.2±7.1±11.3 MeV/c2 and a width of Γ=139.8±12.3±20.6 MeV, where the first uncertainties are statistical and the second ones are systematic. In addition, the timelike electromagnetic form factor of the kaon is determined at the individual center-of-mass energy points.