Knowing the lengthscales at which turbulent fluctuations dissipate is key to understanding the nature of weakly compressible magnetohydrodynamic turbulence. We use radio wavelength interferometric imaging observations which measure the extent to which distant cosmic sources observed against the inner solar wind are scatter-broadened. We interpret these observations to determine that the dissipation scales of solar wind density turbulence at heliocentric distances of 2.5 -- 20.27 $R_{odot}$ range from $approx$ 13500 to 520 m. Our estimates from $approx$ 10--20 $R_{odot}$ suggest that the dissipation scale corresponds to the proton gyroradius. They are relevant to in-situ observations to be made by the Parker Solar Probe, and are expected to enhance our understanding of solar wind acceleration.