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We examine HD 207651 as a possible example of a star exhibiting both γ Doradus and δ Scuti type pulsations. We find photometric periods of 0.06479 and 0.06337 days with peak-to-peak amplitudes in Johnson B of 21 and 13 mmag, respectively, clearly indicating δ Scuti pulsations. Additional light variation with a period of 0.73540 days and an even larger amplitude of 31 mmag is within the range of γ Doradus pulsation periods but results instead from the ellipticity effect. HD 207651 has a composite spectrum with a weak, narrow absorption line superposed near the center of each broad metal line. The broad-lined component is the primary of a short-period, single-lined binary, which has a period of 1.4708 days, twice the period of the ellipsoidal variations seen in the photometry. We determine the primary to be an A8 giant and estimate the unseen secondary of the short-period binary to be a mid-M dwarf. The narrow-lined star, an F7: dwarf, shows velocity variability with a period of months or perhaps years. It is thus a more distant companion to the binary, making HD 207651 a triple system. All light variations come from the A8 giant primary star. Since the 0.73540 day variation results from the ellipticity effect, HD 207651 is not an example of a star that exhibits both δ Scuti and γ Doradus pulsations. The growing number of confirmed γ Doradus stars that also occur within the δ Scuti instability strip but fail to show additional δ Scuti variability makes it increasingly unlikely that the two types of pulsation can coexist in the same star.