The spatial distribution of the emission in the v = 1-0 S(1) line of H2 in the planetary nebula NGC 7027 is presented. The excited H2 molecules do not fill the same volume as the ionized gas and most likely reside near the outer edge of the nebula. The spatial resolution of the data, 5 arcsec, is insufficient to define the location of the molecules precisely. An upper limit to the strength of the v = 2-1 S(1) line is given; it is low enough to preclude simple ultraviolet fluorescence as the source of excitation. A simple shock model can fit the line ratio data, however, and there is enough energy in the expanding nebula to sustain such a shock. A rough estimate of 1 M_⊙ to 4 M_⊙ for the mass in the molecular cloud surrounding NGC 7027 is derived
