  The renormalised value of $<\phi^2>$ is calculated for a massless,
conformally coupled scalar field in the Hartle-Hawking vacuum state. This
calculation is a first step towards the calculation of the gravitational back
reaction of the field in a black cosmic string spacetime which is
asymptotically anti-DeSitter and possesses a non constant dilaton field. It is
found that the field is divergence free throughout the spacetime and attains
its maximum value near the horizon.
