  The NA48/2 experiment at CERN reports the first observation of the $K^{\pm}
\rightarrow \pi^{0} \pi^{0} \mu^{\pm} \nu$ decay based on a sample of 2437
candidates with 15% background contamination collected in 2003--2004. The decay
branching ratio in the kinematic region of the squared dilepton mass above
$0.03$~GeV$^2/c^4$ is measured to be $(0.65 \pm 0.03) \times 10^{-6}$. The
extrapolation to the full kinematic space, using a specific model, is found to
be $(3.45 \pm 0.16) \times 10^{-6}$, in agreement with chiral perturbation
theory predictions.
