We combine long-term photometric observations in multiple band passes to determine the rotational light curve for the binary Kuiper-Belt object (79360) Sila-Nunam. We measure an unambiguous fundamental period of 6.2562 ± 0.002 d, within 0.02% of half the orbital period (Porb = 12.50995 ± 0.00036 d) determined earlier from HST observations resolving the binary. The light curve is double-peaked, and well fit by the sum of two sinusoids: a primary with period Porb/2 and peak-to-peak amplitude 0.120 ± 0.012 mag and a secondary with period Porb and peak-to-peak amplitude 0.044 ± 0.010 mag. Excluding observations within ∼0.1 deg of opposition, we measure a linear solar phase dependence with slope 0.147 ± 0.018 mag deg-1 and a mean absolute magnitude in the Gunn g band of 6.100 ± 0.006 mag. There is no rotational color variation exceeding 4%. We also observe that eclipses occur centered on light curve minima to within 0.3%, requiring the long axis of at least one of the two bodies to point precisely toward the other. Assuming the binary is doubly synchronous and both rotation axes are aligned with the orbital angular momentum vector, our observations jointly constrain triaxial shape models for Sila and Nunam such that the product of their long-to-intermediate axes ratios is 1.120 ± 0.01. Hence both bodies are elongated by 6%, or else one is elongated by 6-12%, and the other by less than 6%.