Weak gravitational lensing has the potential to place tight constraints on the equation of the state of dark energy. However, this will only be possible if shear measurement methods can reach the required level of accuracy. We present a new method for measuring the ellipticity of galaxies used in weak lensing surveys. The method makes use of direct deconvolution of the data by the total point spread function (PSF). We adopt a linear algebra formalism that represents the PSF as a Toeplitz matrix. This allows us to solve the convolution equation by applying the Hopfield neural network iterative scheme. The ellipticity of galaxies in the deconvolved images are then measured using second-order moments of the autocorrelation function of the images. To our knowledge, it is the first time full image deconvolution has been used to measure weak lensing shear. We apply our method to the simulated weak lensing data proposed in the GREAT10 challenge and obtain a quality factor of Q = 87. This result is obtained after applying image denoising to the data, prior to the deconvolution. The additive and multiplicative biases on the shear power spectrum are then root A = +0.09 x 10(-4) and M/2 = +0.0357, respectively.