  Precision weak gravitational lensing experiments require measurements of
galaxy shapes accurate to <1 part in 1000. We investigate measurement biases,
noted by Voigt and Bridle (2009) and Melchior et al. (2009), that are common to
shape measurement methodologies that rely upon fitting elliptical-isophote
galaxy models to observed data. The first bias arises when the true galaxy
shapes do not match the models being fit. We show that this "underfitting bias"
is due, at root, to these methods' attempts to use information at high spatial
frequencies that has been destroyed by the convolution with the point-spread
function (PSF) and/or by sampling. We propose a new shape-measurement technique
that is explicitly confined to observable regions of k-space. A second bias
arises for galaxies whose ellipticity varies with radius. For most
shape-measurement methods, such galaxies are subject to "ellipticity gradient
bias". We show how to reduce such biases by factors of 20--100 within the new
shape-measurement method. The resulting shear estimator has multiplicative
errors <1 part in 1000 for high-S/N images, even for highly asymmetric
galaxies. Without any training or recalibration, the new method obtains Q=3000
in the GREAT08 Challenge of blind shear reconstruction on low-noise galaxies,
several times better than any previous method.
