In this paper, we establish a certain L-infinity-null controllability for the internally controlled heat equation in Omega x [0, T], with the control restricted to a product set of an open nonempty subset in Omega and a subset of positive measure in the interval [0, T]. Based on this, we obtain a bang-bang principle for the time optimal control of the heat equation with controls taken from the set U-ad = {u(., t) : [0, infinity)-> L-2(Omega) measurable; u(., t) is an element of U, a. e. in t}, where U is a closed and bounded subset of L-2(Omega). Namely, each optimal control u*(., t) of the problem satisfies necessarily the bang-bang property: u*(., t) is an element of partial derivative U for almost all t is an element of [0, T*], where partial derivative U denotes the boundary of the set U and T* is the optimal time. We also get the uniqueness of the optimal control when the target set S is convex and the control set U is a closed ball.