The neutron-rich nucleus Ba-144 (t(1/2) = 11.5 s) is expected to exhibit some of the strongest octupole correlations among nuclei with mass numbers A less than 200. Until now, indirect evidence for such strong correlations has been inferred from observations such as enhanced E1 transitions and interleaving positive- and negative-parity levels in the ground-state band. In this experiment, the octupole strength was measured directly by sub-barrier, multistep Coulomb excitation of a post-accelerated 650-MeV Ba-144 beam on a 1.0-mg/cm(2) Pb-208 target. The measured value of the matrix element, < 3(1)(-)parallel to M(E3)parallel to 0(1)(+)> = 0.65((+17)(-23)) eb(3/2,) corresponds to a reduced Bd(E3) transition probability of 48((+25)(-34)) W.u. This result represents an unambiguous determination of the octupole collectivity, is larger than any available theoretical prediction, and is consistent with octupole deformation.