We report the determination of the Dzyaloshinsky-Moriya interaction, the dominant magnetic anisotropy term in the kagome spin-1/2 compound ZnCu3(OH)6Cl2. Based on the analysis of the high-temperature electron spin resonance (ESR) spectra, we find its main component |Dz|=15(1) K to be perpendicular to the kagome planes. Through the temperature dependent ESR linewidth, we observe a building up of nearest-neighbor spin-spin correlations below approximately 150 K.