The radioactive radium-225 (225Ra) atom is a favorable system to search for a permanent electric dipole moment (EDM). Due to its strong nuclear octupole deformation and large atomic mass, 225Ra is particularly sensitive to interactions in the nuclear medium that violate both time-reversal symmetry and parity. We have developed a cold-atom technique to study the spin precession of 225Ra atoms held in an optical dipole trap, and demonstrated the principle of this method by completing the first measurement of its atomic EDM, reaching an upper limit of |d(225Ra)|< 4.3 x 10^-22 ecm (95% confidence).