(Joint work with Leonid Mytnik). It is well known from the papers of Zvonkin-Veretennikov-Davie-Flandoli that ordinary differential equations (ODEs) regularize in the presence of noise. Even if an ODE is “very bad” and has no solutions (or has many solutions), then the addition of a random noise leads almost surely to a “nice” ODE with a unique solution. We investigate the same phenomenon for a 1D heat equation with an irregular drift. We prove existence and uniqueness of the flow of solutions and, as a byproduct of our proof, we also establish path-by-path uniqueness. This extends recent results of Davie (2007) to the context of stochastic partial differential equations.