  We examine the linear stability of static, spherically symmetric wormhole
solutions of Einstein's field equations coupled to a massless ghost scalar
field. These solutions are parametrized by the areal radius of their throat and
the product of the masses at their asymptotically flat ends. We prove that all
these solutions are unstable with respect to linear fluctuations and possess
precisely one unstable, exponentially in time growing mode. The associated time
scale is shown to be of the order of the wormhole throat divided by the speed
of light. The nonlinear evolution is analyzed in a subsequent article.
