The initial-final mass relation represents a mapping between the mass of a white dwarf remnant and the mass that the hydrogen-burning main-sequence star that created it once had. The empirical relation thus far has been constrained using a sample of image40 stars in young open clusters, ranging in initial mass from image2.75 to 7 image, and shows a general trend that connects higher mass main-sequence stars with higher mass white dwarfs. In this paper, we present CFHT CFH12K photometric and Keck LRIS multiobject spectroscopic observations of a sample of 22 white dwarfs in two older open clusters, NGC 7789 (image Gyr) and NGC 6819 (image Gyr). We measure masses for the highest signal-to-noise ratio spectra by fitting the Balmer lines to atmosphere models and place the first direct constraints on the low- mass end of the initial-final mass relation. Our results indicate that the observed general trend at higher masses continues down to low masses, with image main-sequence stars forming image white dwarfs. When added to our new data from the very old cluster NGC 6791, the relation is extended down to image (corresponding to image). This extension of the relation represents a fourfold increase in the total number of hydrogen-burning stars for which the integrated mass loss can now be calculated from empirical data, assuming a Salpeter initial mass function. The new leverage at the low-mass end is used to derive a purely empirical initial-final mass relation. The sample of white dwarfs in these clusters also shows several interesting systems that we discuss further: a DB (helium) white dwarf, a magnetic white dwarf, a DAB (mixed hydrogen/helium atmosphere or a double degenerate DA+DB) white dwarf(s), and two possible equal-mass da double degenerate binary systems.