We report shear modulus (G) and internal friction (Q−1) measurements of large-area monolayer graphene films grown by chemical vapor deposition on copper foil and transferred onto high-Q silicon mechanical oscillators. The shear modulus, extracted from a resonance frequency shift at 0.4 K where the apparatus is most sensitive, averages 280 GPa. This is five times larger than those of the multilayered graphene-based films measured previously. The internal friction is unmeasurable within the sensitivity of our experiment and thus bounded above by Q−1 ≤ 3 × 10−5, which is orders-of-magnitude smaller than that of multilayered graphene-based films. Neither annealing nor interface modification has a measurable effect on G or Q−1. Our results on G are consistent with recent theoretical evaluations and simulations carried out in this work, showing that the shear restoring force transitions from interlayer to intralayer interactions as the film thickness approaches one monolayer.