  We propose a method to increase both the neutron storage time and the
precision of its lifetime measurements by at least tenfold. The storage of
ultracold neutrons (UCN) in material traps now provides the most accurate
measurements of neutron lifetime and is used in many other experiments. The
precision of these measurements is limited by the interaction of UCN with the
trap walls. We show that covering trap walls with liquid helium may strongly
decrease the UCN losses from material traps. $^4$He does not absorb neutrons at
all. Superfluid He covers the trap walls as a thin film, $\sim 10$ nm thick,
due to the van der Waals attraction. However, this He film on a flat wall is
too thin to protect the UCN from their absorption inside a trap material. By
combining the van der Waals attraction with capillary effects we show that
surface roughness may increase the thickness of this film much beyond the
neutron penetration depth $\sim 33$nm. Using liquid He for UCN storage requires
low temperature $T<0.5$ K to avoid neutron interaction with He vapor, while the
neutron losses because of the interaction with surface waves are small and can
be accounted for using their linear temperature dependence.
