A calculation of the $B_{B}$ parameter in the static limit

We calculate the $B_{B}$ parameter, relevant for $\overline{B}^0$ -- $B^0$ mixing, from a lattice gauge theory simulation at $\beta = 6.0$. The bottom quarks are simulated in the static theory, the light quarks with Wilson fermions. Improved smearing functions produced by a variational technique, MOST, are used to reduce statistical errors and minimize excited-state contamination of the ground-state signal. We obtain $B_B(4.33 GeV) = 0.98^{+4}_{-4}$ (statistical) $^{+3}_{-18}$ (systematic) which corresponds to $\widehat{B}_B = 1.40^{+6}_{-6}$ (statistical) $^{+4}_{-26}$ (systematic) for the one-loop renormalization-scheme-independent parameter. The systematic errors include the uncertainty due to alternative (less favored) treatments of the perturbatively-calculated mixing coefficients; this uncertainty is at least as large as residual differences between Wilson-static and clover-static results. Our result agrees with extrapolations of results from relativistic (Wilson) heavy quark simulations.