We study the process e+e− --> K+K− with a hard photon radiated from the initial state. About 76,000 fully reconstructed events have been selected from 232 fb−1 of BaBar data. The invariant mass of the hadronic final state defines the effective e+e− center-of-mass energy, so that these data can be compared with the corresponding direct e+e− measurements. From the K+K−-mass spectrum, the cross section for the process e+e− --> K+K− are measured for center-of-mass energies from 0.99 to 5.0 GeV. We observe the (1020), Y (2175) and J/ signals for the kaons pair final state and measure the corresponding branching fractions.