The absolute thermoelectric power of dilute gold-platinum alloys, containing 0.11, 0.50, 1.03, and 4.99 at.% Pt, has been measured between 4.2 and 300\ifmmode^\circ\else\textdegree\fi{}K. Also, the change in the thermoelectric power of the alloys due to quenched-in lattice vacancies has been determined. The alloys containing 1 at.% Pt or less show clearly a positive phonon-drag component ${S}^{g}$ of the thermopower below about 150\ifmmode^\circ\else\textdegree\fi{}K. The positive values of ${S}^{g}$ become smaller with increasing platinum concentration. At 20 to 30\ifmmode^\circ\else\textdegree\fi{}K, ${S}^{g}$ in these alloys changes its sign and is negative at very low temperatures. It is suggested that the appearance of a negative phonon-drag thermopower in the dilute Au-Pt alloys at very low temperatures is based on the anisotropy of the relaxation times for electron scattering. Quenched-in lattice vacancies cause a reduction of ${S}^{g}$ in the alloys containing 1 at.% Pt or less. The reduction of $|{S}^{g}|$ due to vacancies is very pronounced at low temperatures, where ${S}^{g}$ in the annealed alloys is negative. This is explained by the anisotropy of the electron scattering by vacancies, which apparently is opposite to the anisotropy of the electron scattering by the platinum ions, thus leading to an enhancement, due to vacancies, of the positive contribution to ${S}^{g}$ at low temperatures.