  The origin of the observed acceleration of the expansion of the universe is a
major problem of modern cosmology and theoretical physics. Simple estimations
of the contribution of vacuum to the density energy of the universe in quantum
field theory are known to lead to catastrophic large values compared to
observations. Such a contribution is therefore generally not regarded as a
viable source for the acceleration of the expansion. In this letter we propose
that the vacuum contribution actually provides a small positive value to the
density energy of the universe. The underlying mechanism is a manifestation of
the quantum nature of the gravitational field, through a Casimir-like effect
from an additional compact dimension of space. A key ingredient is to assume
that only modes with wavelength shorter than the Hubble length contribute to
the vacuum. Such a contribution gives a positive energy density, has a Lorentz
invariant equation of state in the usual 4D spacetime and hence can be
interpreted as a cosmological constant. Its value agrees with observations for
a radius of a 5th extra dimension given by $35\,\mu$m. This implies a
modification of the gravitational inverse square law around this scale, close
but below existing limits from experiments testing gravity at short range.
