  We present a measurement of the forward-backward asymmetry in the production
of $B^{\pm}$ mesons, $A_{\rm FB}(B^{\pm})$, using $B^{\pm} \rightarrow J/\psi
K^{\pm}$ decays in 10.4 ${\rm fb}^{-1}$ of $p\bar{p}$ collisions at $\sqrt{s} =
1.96$ TeV collected by the D0 experiment during Run II of the Tevatron
collider. A nonzero asymmetry would indicate a preference for a particular
flavor, i.e., $b$ quark or $\bar{b}$ antiquark, to be produced in the direction
of the proton beam. We extract $A_{\rm FB}(B^{\pm})$ from a maximum likelihood
fit to the difference between forward- and backward-produced $B^{\pm}$ mesons.
We measure an asymmetry consistent with zero: $A_{\rm FB}(B^{\pm})$ = [$-$0.24
$\pm$ 0.41 (stat) $\pm$ 0.19 (syst)]%.
