A measurement of the branching ratio of the rare leptonic
kaon decay $K^\pm \rightarrow \mu^\pm \nu_{\mu} e^+ e^- $ is
presented using data collected by the NA48/2 experiment in 2003 and
2004. The measurement is performed in the region $M_{ee} >
140$~MeV/c$^2$. In this particular region low energy QCD
contributions become important and can be calculated in
the framework of Chiral Perturbation Theory (ChPT). From a total number
of $1.56 \times 10^{11}$ recorded kaon decays, the
branching ratio is measured to be $\mathcal{B} \left( K^\pm
\rightarrow \mu^\pm \nu_{\mu} e^+ e^- | M_{ee} > 140~ MeV/c^2
\right) = (7.8 \pm 0.2) \times 10^{-8} $.