We introduce a method for identifying "twin" Type Ia supernovae (SNeIa) and using them to improve distance measurements. This novel approach to SN Ia standardization is made possible by spectrophotometric time series observations from the Nearby Supernova Factory (SNfactory). We begin with a well-measured set of SNe, find pairs whose spectra match well across the entire optical window, and then test whether this leads to a smaller dispersion in their absolute brightnesses. This analysis is completed in a blinded fashion, ensuring that decisions made in implementing the method do not inadvertently bias the result. We find that pairs of SNe with more closely matched spectra indeed have reduced brightness dispersion. We are able to standardize this initial set of SNfactory SNe to 0.083+/-0.012mag, implying a dispersion of 0.072+/-0.010mag in the absence of peculiar velocities. We estimate that with larger numbers of comparison SNe, e.g., using the final SNfactory spectrophotometric data set as a reference, this method will be capable of standardizing high-redshift SNe to within 0.06-0.07mag. These results imply that at least 3/4 of the variance in Hubble residuals in current SN cosmology analyses is due to previously unaccounted-for astrophysical differences among the SNe.