In laser fusion, excessive preheat of the DT fuel by hot electrons reduces the achievable thermonuclear energy gain from the imploded target. A criterion defining the tolerable amount of preheat is derived which implies that the work needed to compress the fuel must not exceed the least possible (minimum entropy) value by more than a factor of about three. – The amount of high-Z shielding that is required to limit fuel preheat to this permissible level is calculated, and the laser-pulse energy WL that is needed to implode an adequately shielded target is then determined. An explicit relation is obtained that describes the dependence of WL on the various physical parameters of the system. – It is concluded that hot-electron preheat may preclude adequate energy gain with targets driven by 1.06-μm light unless pulse energies in excess of one mega-joule are available. It appears, however, that frequency doubling or tripling of 1.06-μm light can reduce hot-electron preheat to a relatively insignificant level, and is an option that is currently receiving serious consideration.