The expectation value of the stress - energy tensor of a free conformally invariant scalar field is computed in a two-dimensional reduction of the Alcubierre `warp-drive' spacetime. Unless the spacetime is in the Hartle - Hawking state at an appropriate temperature, the stress - energy diverges on past and future event horizons which form when the apparent velocity of the spaceship exceeds the speed of light. The likelihood of the spacetime being in this state, whether due to natural evolution or the application of technology, is briefly discussed.