Single SF6 molecules have been embedded in large He-4 droplets ((N) over bar(4) approximate to 10(3) - 10(4) atoms) in a molecular beam and studied via infrared laser depletion spectroscopy. The rotational fine structure of the nu(3) transition is analyzed with an effective third order gas phase Hamiltonian of SF6 yielding seven spectroscopic parameters as a function of the measured mean droplet size. From the intensities of the rotational lines the rotational temperature in droplets with (N) over bar(4) > 10(3) atoms is determined to be T-rot = 0.38(1) K consistent with theoretical estimates. Quantitative information on the efficiency of evaporative cooling of the droplets could be obtained by increasing the droplet temperature up to T-rot = 0.55 K by many successive inelastic collisions with He-4 atoms from the background gas. For small droplets the absorption maximum shows an increasing redshift with respect to the gas phase and only small downward shift with droplet sizes (N) over bar(4) > 2 X 10(3). This could not be explained quantitatively with the excluded volume model assuming a liquid drop nor by the expected decrease in the internal pressure with increasing droplet radius. The much smaller decrease in the redshift observed is shown to provide evidence that the nearest neighbor shell of He-4 next to the molecule is almost incompressible providing direct experimental evidence for the existence of a dense snowball-like shell structure of He-4 atoms around the molecule as predicted theoretically.