  We consider a Casimir apparatus consisting of two perfectly conducting
parallel plates, subject to the weak gravitational field of the Earth. The aim
of this paper is the calculation of the energy-momentum tensor of this system
for a free, real massless scalar field satisfying Neumann boundary conditions
on the plates. The small gravity acceleration (here considered as not varying
between the two plates) allows us to perform all calculations to first order in
this parameter. Some interesting results are found: a correction, depending on
the gravity acceleration, to the well-known Casimir energy and pressure on the
plates. Moreover, this scheme predicts a tiny force in the upwards direction
acting on the apparatus. These results are supported by two consistency checks:
the covariant conservation of the energy-momentum tensor and the vanishing of
its regularized trace, when the scalar field is conformally coupled to gravity.
