  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 relation thus far has been constrained using a
sample of ~40 stars in young open clusters, ranging in initial mass from ~2.75
-- 7 Msun, 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 (t = 1.4 Gyr) and NGC 6819
(t = 2.5 Gyr). We measure masses for the highest S/N 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
M_initial = 1.16 Msun main-sequence stars forming M_final = 0.53 Msun white
dwarfs (including our data from the very old open cluster, NGC 6791). This
extention of the relation represents a four fold increase in the total number
of hydrogen burning stars for which the integrated mass loss can now be
calculated, 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
without the need for any indirectly derived anchor points. The sample of white
dwarfs in these clusters also shows several very interesting systems that we
discuss further: a DB (helium atmosphere) 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.
