We have detected the v = 1 --> 0 S(1) (lambda= 2.1218 mum) and v = 2 --> 1 S(1) (lambda= 2.2477 mum) lines of H-2 in the Galactic Centre, in a 90 x 27 arcsec(2) region between the north-eastern boundary of the non-thermal source Sgr A East, and the giant molecular cloud (GMC) M-0.02 - 0.07. The detected H-2 v = 1 --> 0 S(1) emission has an intensity of 1.6-21 x 10(-18) W m(-2) arcsec(-2) and is present over most of the region. Along with the high intensity, the large linewidths (FWHM = 40-70 km s(-1) ) and the H-2 v = 2 --> 1 S(1) to v = 1 --> 0 S(1) line ratios (0.3-0.5) can be best explained by a combination of C-type shocks and fluorescence. The detection of shocked H-2 is clear evidence that Sgr A East is driving material into the surrounding adjacent cool molecular gas. The H-2 emission lines have two velocity components at similar to+50 and similar to0 km s(-1) , which are also present in the NH3 (3, 3) emission mapped by McGary, Coil & Ho. This two-velocity structure can be explained if Sgr A East is driving C-type shocks into both the GMC M-0.02 - 0.07 and the northern ridge of McGary et al.