In this paper we develop a method for constructing strong solutions of one-dimensional SDE’s where the drift may be discontinuous and unbounded. The driving noise is the Brownian Motion. In addition to existence and uniqueness of the strong solution, we show that the solution is Sobolev-differentiable in the initial condition and Malliavin differentiable. The method is based on Malliavin calculus using a sim- ilar technique as initiated in  and further developed in  and  where the authors consider bounded coefficients. This method is not based on a pathwise uniqueness argument. We will apply these results to the stochastic transport equation. More specifically, we ob- tain a continuously differentiable solution of the stochastic transport equation when the driving function is a step function.