  A linear stochastic vector advection equation is considered; the equation may
model a passive magnetic field in a random fluid. When the driving velocity
field is rough but deterministic, in particular just H\"{o}lder continuous and
bounded, one can construct examples of infinite stretching of the passive
field, arising from smooth initial conditions. The purpose of the paper is to
prove that infinite stretching is prevented if the driving velocity field
contains in addition a white noise component.
