  We have measured the isospin mixing of the I$^\pi$= 1/2$^+$ E$_x$=2.599 MeV
state in nearly-doubly-magic $^{47}$Ca with the isobaric analog 1/2$^+$ state
of $^{47}$K. Using the TRIUMF atom trap for $\beta$ decay, we have measured a
nonzero asymmetry of the progeny $^{47}$Ca with respect to the initial $^{47}$K
spin polarization, which together with the $\beta$ asymmetry implies a nonzero
ratio of Fermi to Gamow-Teller matrix elements $y$= 0.098$\pm$0.037 for the
$1/2^+ \rightarrow 1/2^+$ transition. Interpreting $y$ as mixing between this
state and the isobaric analog state implies a Coulomb matrix element magnitude
101 $\pm$ 37 keV. This relatively large matrix element supports a model from
the literature of analog-antianalog isospin mixing, which predicts large matrix
elements in cases involving excess neutrons over protons occupying more than
one major shell. The result supports pursuing a search for time-reversal odd,
parity-even, isovector interactions using a correlation in $^{47}$K $\beta$
decay.
