  We calculate the ground state energy of a massive scalar field in the
background of a cosmic string of finite thickness (Gott-Hiscock metric). Using
zeta functional regularization we discuss the renormalization and the relevant
heat kernel coefficients in detail. The finite (non local) part of the ground
state energy is calculated in 2+1 dimensions in the approximation of a small
mass density of the string. By a numerical calculation it is shown to vanish as
a function of the radius of the string and of the parameter $\xi$ of the
nonconformal coupling.
