  The cross section of the process $e^{+} e^{-} \rightarrow K^{+} K^{-}$ is
measured at a number of center-of-mass energies $\sqrt{s}$ 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 $M=2239.2 \pm 7.1 \pm 11.3$~and a width of
$\Gamma=139.8\pm12.3\pm20.6$ MeV, where the first uncertainties are statistical
and the second ones are systematic. In addition, the time-like electromagnetic
form factor of the kaon is determined at the individual center-of-mass energy
points.
