The most commonly used index of stellar magnetic activity is the instrumental flux scale of singly ionized calcium H & K line core emission, S, developed by the Mount Wilson Observatory (MWO) HK Project, or the derivative index R'_HK_. Accurately placing the Sun on the S scale is important for comparing solar activity to that of the Sun-like stars. We present previously unpublished measurements of the reflected sunlight from the Moon using the second-generation MWO HK photometer during solar cycle 23 and determine cycle minimum S_23,min_=0.1634+/-0.0008, amplitude {Delta}S_23_=0.0143+/-0.0012, and mean <S_23_>=0.1701+/-0.0005. By establishing a proxy relationship with the closely related National Solar Observatory Sacramento Peak calcium K emission index, itself well correlated with the Kodaikanal Observatory plage index, we extend the MWO S time series to cover cycles 15-24 and find on average <S_min_>=0.1621+/-0.0008, <{Delta}S_cyc_>=0.0145+/-0.0012, <S_cyc_>=0.1694+/-0.0005. Our measurements represent an improvement over previous estimates that relied on stellar measurements or solar proxies with non-overlapping time series. We find good agreement from these results with measurements by the Solar-Stellar Spectrograph at Lowell Observatory, an independently calibrated instrument, which gives us additional confidence that we have accurately placed the Sun on the S-index flux scale.