  We report the observation and analysis of the $3\,^3\Pi_{\Omega}$
double-minimum electronic excited state of $^{39}$K$^{85}$Rb. The spin-orbit
components ($0^{+}, 0^{-}, 1$ and 2) of this state are investigated based on
potentials developed from the available \emph{ab initio} potential curves. We
have assigned the vibrational levels $v'=2-11$ of the $3\,^3\Pi_{1,2}$
potentials and $v'=2-12$ of the $3\,^3\Pi_{0^{+/-}}$ potential. We compare our
experimental observations of the $3\,^3\Pi_{\Omega}$ state with predictions
based on theoretical potentials. The observations are based on resonance
enhanced multiphoton ionization (REMPI) of ultracold KRb in vibrational levels
$v"=14-25$ of the $a\,^3\Sigma^+$ state. These \emph{a}-state ultracold
molecules are formed by photoassociation of ultracold $^{39}$K and $^{85}$Rb
atoms to the 5(1) state of KRb followed by spontaneous emission to the \emph{a}
state.
