  During the first half of their main-sequence lifetimes, stars rapidly lose
angular momentum to their magnetized winds, a process known as magnetic
braking. Recent observations suggest a substantial decrease in the magnetic
braking efficiency when stars reach a critical value of the Rossby number, the
stellar rotation period normalized by the convective overturn timescale. Cooler
stars have deeper convection zones with longer overturn times, reaching this
critical Rossby number at slower rotation rates. The nature and timing of the
transition to weakened magnetic braking has previously been constrained by
several solar analogs and two slightly hotter stars. In this Letter, we derive
the first direct constraints from stars cooler than the Sun. We present new
spectropolarimetry of the old G8 dwarf $\tau$ Cet from the Large Binocular
Telescope, and we reanalyze a published Zeeman Doppler image of the younger G8
star 61 UMa, yielding the large-scale magnetic field strengths and
morphologies. We estimate mass-loss rates using archival X-ray observations and
inferences from Ly$\alpha$ measurements, and we adopt other stellar properties
from asteroseismology and spectral energy distribution fitting. The resulting
calculations of the wind braking torque demonstrate that the rate of angular
momentum loss drops by a factor of 300 between the ages of these two stars
(1.4-9 Gyr), well above theoretical expectations. We summarize the available
data to help constrain the value of the critical Rossby number, and we identify
a new signature of the long-period detection edge in recent measurements from
the Kepler mission.
