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Understanding the physics of turbulence is crucial for many applications, including weather, industry, and astrophysics. In the interstellar medium (ISM), supersonic turbulence plays a crucial role in controlling the gas density and velocity structure, and ultimately the birth of stars. Here we present a simulation of interstellar turbulence with a grid resolution of 10048^3 cells that allows us to determine the position and width of the sonic scale (l_s) - the transition from supersonic to subsonic turbulence. The simulation simultaneously resolves the supersonic and subsonic cascade, v(l) ~ l^p, where we measure p_sup = 0.49 +/- 0.01 and p_sub = 0.39 +/- 0.02, respectively. We find that l_s agrees with the relation l_s / L = phi_s Mach^(-1/p_sup), where Mach is the three-dimensional Mach number, and L is either the driving scale of turbulence or the diameter of a molecular cloud. If L is the driving scale, we measure phi_s = 0.42 (+0.12) (-0.09), primarily because of the separation between the driving scale and the start of the supersonic cascade. For a supersonic cascade extending beyond the cloud scale, we get phi_s = 0.91 (+0.25) (-0.20). In both cases, phi_s < 1, because we find that the supersonic cascade transitions smoothly to the subsonic cascade over a factor of 3 in scale, instead of a sharp transition. Our measurements provide quantitative input for turbulence-regulated models of filament structure and star formation in molecular clouds.