The three-dimensional, time-dependent temperature distribution in a moving solid of finite thickness due to a laser beam with a Gaussian power distribution on its surface is investigated. The effects due to latent heat and the keyhole are neglected. The model is determined by Peclet number Pe=r0U/2 kappa , a non-dimensional melting temperature Theta m= lambda r0 pi 32/(Tm-T0)/P and a non-dimensional thickness xi 0=h/r0. An upper limit for the time needed to establish the steady state is 0.1 s in the case of iron for all travel speeds. Accounting for the effects due to the finite thickness of the specimen is essential for the thin metal sheets used in high-speed laser welding. Asymptotic solutions for high Pe are provided. The resulting weld pool for high Pe are long, narrow and shallow; the weld pool may be approximated by a cylinder. For a given value of the power, the weld pool length depends only slightly on Pe, and consequently a simple approximate formula for the dependence on laser power P is possible and is presented.