Photometric observations of Neptune with a highly accurate doubly differential photometer, two filters centered in the 6190-A band of methane, and two immediately adjacent continuum filters centered at 5993 and 6332 A are reported which were performed in the interval from March 29 to August 8, 1977. The principal rotation period of at least a major methane albedo feature of Neptune's deeper atmosphere is found to be 18.44 + or - 0.01 (est. error) hr, assuming that such features are stable in longitude over weeks or months. The result obtained is compared with previous determinations, and reasons for a discrepancy between the present and a previous photometric period are considered. It is suggested that the period of 18.44 hr is likely to be quite close to that of the main body of Neptune.