  We consider the Casimir interaction between a cylinder and a hollow cylinder,
both conducting, with parallel axis and slightly different radii. The Casimir
force, which vanishes in the coaxial situation, is evaluated for both small and
large eccentricities using the proximity approximation. The cylindrical
configuration offers various experimental advantages with respect to the
parallel planes or the plane-sphere geometries, leading to favourable
conditions for the search of extra-gravitational forces in the micrometer range
and for the observation of finite temperature corrections.
