In addition to the well known-sign problem, methods currently used to study finite baryon density lattice QCD suffer from additional uncontrolled systematics, coming e.g. from the analytic continuation problem one faces with the Taylor or imaginary chemical potential methods. We formulate and test a new method - sign reweighting - that works directly at finite chemical potential and is free from any such uncontrolled systematics. In particular it is free of the overlap problem, which bottlenecked more traditional reweighting methods so far. With this approach the only problem is the sign problem itself. As a first test, we apply the method to calculate the position of the critical endpoint on a coarse lattice: unimproved staggered fermions at $N_\tau=4$; as a second application, we also study the phase diagram with 2stout improved staggered fermions at $N_\tau=6$. This second one is already a reasonably fine lattice - relevant for phenomenology.