  Approaches to finite baryon density lattice QCD usually suffer from
uncontrolled systematic uncertainties in addition to the well-known sign
problem. We test a method - sign reweighting - that works directly at finite
chemical potential and is yet free from any such uncontrolled systematics: with
this approach the only problem is the sign problem itself. In practice the
approach involves the generation of configurations with the positive fermionic
weights given by the absolute value of the real part of the quark determinant,
and a reweighting by a sign. There are only two sectors, +1 and -1 and as long
as the average $\left\langle \pm \right\rangle \neq 0$ (with respect to the
positive weight) this discrete reweighting has no overlap problem - unlike
reweighting from $\mu=0$ - and the results are reliable. We also present
results based on this algorithm on the phase diagram of lattice QCD with two
different actions: as a first test, we apply the method to calculate the
position of the critical endpoint with unimproved staggered fermions at
$N_\tau=4$; as a second application, we 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. We demonstrate that the
method penetrates the region of the phase diagram where the Taylor and
imaginary chemical potential methods lose predictive power.
