  This paper reports on a new high precision measurement of the form factors of
the K_{L}\to \pi^{\pm} \mu^{\mp} \nu_{\mu} decay. The data sample of about
2.3$\times 10^{6}$ events was recorded in 1999 by the NA48 experiment at CERN.
Studying the Dalitz plot density we measured a linear, $\lambda^{'}_{+} =
(20.5\pm 2.2_{stat} \pm 2.4_{syst})\times 10^{-3}$, and a quadratic,
$\lambda^{''}_{+} = (2.6\pm 0.9_{stat} \pm 1.0_{syst})\times 10^{-3}$ term in
the power expansion of the vector form factor. No evidence was found for a
second order term for the scalar form factor; the linear slope was determined
to be $\lambda_{0} = (9.5\pm 1.1_{stat} \pm 0.8_{syst})\times 10^{-3}$. Using a
linear fit our results were: $\lambda_{+} = (26.7\pm 0.6_{stat} \pm 0.8_{syst}
)\times 10^{-3}$ and, $\lambda_{0} = (11.7\pm 0.7_{stat} \pm 1.0_{syst})\times
10^{-3}$. A pole fit of the form factors yields:
  $m_V = (905 \pm 9_{stat} \pm 17_{syst})$ MeV/c$^2$ and
  $m_S = (1400 \pm 46_{stat} \pm 53_{syst})$ MeV/c$^2$.
