  We consider one-dimensional biased voter models, where 1's replace 0's at a
faster rate than the other way round, started in a Heaviside initial state
describing the interface between two infinite populations of 0's and 1's. In
the limit of weak bias, for a diffusively rescaled process, we consider a
measure-valued process describing the local fraction of type 1 sites as a
function of time. Under a finite second moment condition on the rates, we show
that in the diffusive scaling limit there is a drifted Brownian path with the
property that all but a vanishingly small fraction of the sites on the left
(resp. right) of this path are of type 0 (resp. 1). This extends known results
for unbiased voter models. Our proofs depend crucially on recent results about
interface tightness for biased voter models.
