The Coulomb excitation of a thick 24Mg target was studied with 35Cl ions of 61, 57, and 52 MeV. The absolute cross section and the anisotropy of the angular distribution of the ground state transition from the 1.369-MeV state in 24Mg was measured, and their dependence on B(E2) and the quadrupole moment, Q, of the 1.369-MeV state was calculated. A B(E2) of 24.5 ± 2.2 Weisskopf units (W.u.) was deduced from the line shape of the 1.369-MeV gamma ray observed with a 40 cm3 Ge(Li) detector. The quadrupole moment determined from the anisotropy measurement was Q = −0.38 ± 0.16 b. From the cross-section measurement, Q = −0.47 ± 0.19 b was obtained using B(E2) = 24.5 ± 2.2 W.u. The dependence of this value of Q on B(E2) is discussed.