  A sample of 65210 $\mathrm{K^{\pm}} \rightarrow \pi^{0} \pi^{0}
\mathrm{e^{\pm}} \nu$ ($\mathrm{K}_{\mathrm{e}4}^{00}$) decay candidates with
$1\%$ background contamination has been collected in 2003$-$2004 by the NA48/2
collaboration at the CERN SPS. A study of the differential rate provides the
first measurement of the hadronic form factor variation in the plane
$(M_{\pi\pi}^2,M_{e\nu}^2)$ and brings evidence for a cusp-like structure in
the distribution of the squared $\pi^{0} \pi^{0}$ invariant mass around
$4m_{\pi^+}^2$. Exploiting a model independent description of this form factor,
the branching ratio, inclusive of radiative decays, is obtained using the
$\mathrm{K^{\pm}} \rightarrow \pi^{0} \pi^{0} \pi^{\pm}$ decay mode as
normalization. It is measured to be BR($\mathrm{K}_{\mathrm{e}4}^{00}$) =
$(2.552 \pm 0.010_{stat} \pm 0.010_{syst} \pm 0.032_{ext} ) \times 10^{-5}$,
which improves the current world average precision by an order of magnitude
while the $1.4\%$ relative precision is dominated by the external uncertainty
from the normalization mode. A comparison with the properties of the
corresponding mode involving a $\pi^{+} \pi^{-}$ pair
($\mathrm{K}_{\mathrm{e}4}^{+-}$) is also presented.
