  Using a data sample of 921.9 fb$^{-1}$ collected with the Belle detector, we
study the process of $e^+e^-\to D^+_sD_{s1}(2536)^-+c.c.$ via initial-state
radiation. We report the first observation of a vector charmoniumlike state
decaying to $D^+_sD_{s1}(2536)^-+c.c.$ with a significance of 5.9$\sigma$,
including the systematic uncertainties. The measured mass and width are
$(4625.9^{+6.2}_{-6.0}({\rm stat.})\pm0.4({\rm syst.}))~{\rm MeV}/c^{2}$ and
$(49.8^{+13.9}_{-11.5}({\rm stat.})\pm4.0({\rm syst.}))~{\rm MeV}$,
respectively. The product of the $e^+e^-\to D^+_sD_{s1}(2536)^-+c.c.$ cross
section and the branching fraction of $D_{s1}(2536)^-\to{\bar D}^{*0}K^-$ is
measured from the $D_s \bar{D}_{s1}(2536)$ threshold to 5.59~GeV.
