The thermopower of the organic superconductor \ensuremath{\kappa}-(BEDT-TTF${)}_{2}$Cu[N(CN${)}_{2}$]Br single crystals has been measured in two crystallographic directions a and c within the most conducting organic donor molecule plane. [Here BEDT-TTF represents bis(ethylenethio)-tetrathiafulvalene.] While the thermopower in the a direction is positive, the thermopower in the c direction is negative. The drastic anisotropy in thermopower reveals that the carriers in the a direction are holelike, whereas the carriers in the c direction are electronlike. A calculation based on the tight-binding electronic band structure is able to describe the temperature dependence of the anisotropic thermopower, but with a much reduced band dispersion.