In this Letter, 21 cm intensity maps acquired at the Green Bank Telescope are cross-correlated with large-scale structure traced by galaxies in the WiggleZ Dark Energy Survey. The data span the redshift range 0.6 < z < 1 over two fields totaling ~41 deg. sq. and 190 hr of radio integration time. The cross-correlation constrains Omega sub(H I)b sub(H I)r = 0.43 + or - 0.07(stat.) + or - 0.04(sys.) x 10 super(?3), where Omega sub(H I) is the neutral hydrogen (H I) fraction, r is the galaxy-hydrogen correlation coefficient, and b sub(H I) is the H I bias parameter. This is the most precise constraint on neutral hydrogen density fluctuations in a challenging redshift range. Our measurement improves the previous 21 cm cross-correlation at z ~ 0.8 both in its precision and in the range of scales probed.