  We study the Casimir effect for a bosonic string extended between D-branes,
and living in a flat space with an antisymmetric background B-field. We find
the Casimir energy as a function of theB-field, and the mass-parameter of the
string, and accordingly we obtain a B-dependence correction term to the
ground-state mass of the string. We show that for sufficiently large B-field,
the ground state of the string contains real (i.e. non-Tachyonic) particles.
