We describe a technique that for the first time measures star formation rates (SFRs) in damped Lyα systems (DLAs) directly. We assume that massive stars form in DLAs and that the far-ultraviolet (FUV) radiation they emit heats the gas by the grain photoelectric mechanism. We infer the heating rate by equating it to the cooling rate measured by the strength of C II* λ1335.7 absorption. Since the heating rate is proportional to the product of the dust-to-gas ratio, the grain photoelectric heating efficiency, and the SFR per unit area, , we can deduce for DLAs in which the cooling rate and dust-to-gas ratio have been measured. We consider models in which the dust consists of carbonaceous grains and silicate grains. We present two-phase models in which the cold neutral medium (CNM) and warm neutral medium (WNM) are in pressure equilibrium. In the CNM model the line of sight passes through CNM and WNM gas, while in the WNM model the line of sight passes only through WNM gas. Since the grain photoelectric heating efficiency is at least an order of magnitude higher in the CNM than in the WNM, most of the C II* absorption arises in the CNM in the CNM model. We use the measured C II* absorption lines to derive for a sample of ≈30 DLAs in which κ has been inferred from element depletion patterns. We show that the inferred corresponds to an average over the star-forming volume of the DLA rather than to local star formation along the line of sight. We obtain the average and show that = 10^-2.2 M yr^-1 kpc^-2 for the CNM solution and = 10^-1.3 M yr^-1 kpc^-2 for the WNM solution. Interestingly, the SFR per unit area in the CNM solution is similar to that measured in the Milky Way interstellar medium.