Critical experimental studies have been carried out in capillary samples to examine the stability of eutectic morphology in 3D under diffusive growth conditions. Current theoretical models on the stability of eutectic with respect to the spacing are based on 2D models only, and significantly different results are observed for experiments in three-dimensional samples. Through experimental studies of steady-state eutectic growth under diffusive growth conditions in the well-characterized Al-Cu system, it is shown that the range of stable spacing is reduced significantly, and the ratio of the maximum to minimum eutectic spacing in 3D is found to be only 1.2 compared to the predicted 2.0 in 2D. The minimum spacing is found to agree with the theoretical model of Jackson and Hunt. The narrow range of stable spacing is shown to be due to the additional degree of freedom in 3D that gives rise to lamellar creation or elimination in the third dimension. A new mechanism of lamellar creation in the third dimension is observed in which lamellae with spacing values larger than some critical value form sideways perturbations that coalesce at the leading edge of the perturbations and then get detached from the parent lamellae to form new lamellae.