The effect of external turbulence completely modifies how the velocity deficit behind a rigid sphere decays with distance, changing it from an x−1 (or x−2/3 depending on the Reynolds number characterising the flow past the sphere) to x−2 decay, where x is the distance downstream. The wake width is observed to grow linearly with x. This striking change has been reported by three recent studies. We develop a new model to explain these studies which combines a stochastic model for passive wake spreading in a turbulent flow with an integral constraint based on momentum flux conservation. These results also show that, in ambient flow with an integral scale L and intensity It, a new flow regime will ultimately emerge at a distance L/It downstream, where turbulence disperses the velocity deficit with a new x−1 law.