  Statistical weak lensing by large-scale structure -- cosmic shear -- is a
promising cosmological tool, which has motivated the design of several large
upcoming surveys. Here, we present a measurement of cosmic shear using coadded
Sloan Digital Sky Survey (SDSS) imaging in 168 square degrees of the equatorial
region, with r<23.5 and i<22.5, a source number density of 2.2 galaxies per
square arcminute and median redshift of 0.52. These coadds were generated using
a new method described in the companion Paper I that was intended to minimise
systematic errors in the lensing measurement due to coherent PSF anisotropies
that are otherwise prevalent in the SDSS imaging data. We present measurements
of cosmic shear out to angular separations of 2 degrees, along with systematics
tests that (combined with those from Paper I on the catalogue generation)
demonstrate that our results are dominated by statistical rather than
systematic errors. Assuming a cosmological model corresponding to WMAP7 and
allowing only the amplitude of matter fluctuations to vary, we find a best-fit
value of sigma_8=0.636 +0.109 -0.154 (1-sigma); without systematic errors this
would be sigma_8=0.636 +0.099 -0.137 (1-sigma). Assuming a flat LCDM model, the
combined constraints with WMAP7 are sigma_8=0.784 +0.028 -0.026 (1-sigma),
+0.055 -0.054 (2-sigma) and Omega_m h^2=0.1303 +0.0047 -0.0048 (1-sigma)+0.009
-0.009 (2-sigma); the 2-sigma error ranges are respectively 14 and 17 per cent
smaller than WMAP7 alone. Aside from the intrinsic value of such cosmological
constraints from the growth of structure, we identify some important lessons
for upcoming surveys that may face similar issues when combining multi-epoch
data to measure cosmic shear.
