The growth kinetics for individual islands during electrodeposition of copper have been studied using in situ transmission electron microscopy. We show that for sufficiently large overpotentials, the growth kinetics approach the rate laws expected for diffusion-limited growth of hemispherical islands, characterized by two distinct regimes. At short times, the island growth exponent is 0.5 as expected for diffusion-limited growth of uncoupled hemispherical islands, while at longer times, the growth exponent approaches 1/6 as expected for planar diffusion to the growing islands. These results provide the first direct measurements of the growth of individual islands during electrochemical deposition. However, quantitative comparison with rate laws shows that the island radii are smaller than predicted and the island densities are much larger than predicted, processes which are not included and we suggest that this is related to adatom formation and surface diffusion. in conventional growth models.