  The Casimir force is calculated in the configuration of a spherical lens and
a disc of finite radius covered by $Cu$ and $Au$ thin layers which was used in
a recent experiment. The correction to the Casimir force due to finiteness of
the disc radius is shown to be negligible. Also the corrections are discussed
due to the finite conductivity, large-scale and short-scale deviations from the
perfect shape of the bounding surfaces and the temperature correction. They
were found to be essential when confronting the theoretical results with
experimental data. Both Yukawa-type and power-law hypothetical forces are
computed which may act in the configuration under consideration due to the
exchange of light and/or massless elementary particles between the atoms of the
lens and the disc. New constraints on the constants of these forces are
determined which follow from the fact that they were not observed within the
limits of experimental errors. For Yukawa-type forces the new constraints are
up to 30 times stronger than the best ones known up today. A possible
improvement of experimental parameters is proposed which gives the possibility
to strengthen constraints on Yukawa-type interactions up to $10^4$ times and on
power-law interactions up to several hundred times.
