We have experimentally identified the exchange-induced spin blockade in a GaAs double quantum dot. The transport is suppressed only when the eigenstates are well-defined singlet and triplet states, and thus sensitive to dynamic nuclear-spin polarization that causes singlet-triplet mixing. This gives rise to unusual current spectra, such as a sharp current dip and an asymmetric current profile near the triplet resonance of a double quantum dot. Numerical simulations suggest that the current dip is a signature of identical nuclear-spin polarization in the two dots, which is attractive for coherent spin manipulations in a material with nuclear spins.