  All approaches currently used to study finite baryon density lattice QCD
suffer from uncontrolled systematic uncertainties in addition to the well-known
sign problem. We formulate and test an algorithm, sign reweighting, that works
directly at finite $\mu = \mu_B/3$ and is yet free from any such uncontrolled
systematics. With this algorithm the {\em only} problem is the sign problem
itself. This approach involves the generation of configurations with the
positive fermionic weight $|{\rm Re\; det} D(\mu)|$ where $D(\mu)$ is the Dirac
matrix and the signs ${\rm sign} \; ( {\rm Re\; det} D(\mu) ) = \pm 1$ are
handled by a discrete reweighting. Hence there are only two sectors, $+1$ and
$-1$ and as long as the average $\langle\pm 1\rangle \neq 0$ (with respect to
the positive weight) this discrete reweighting by the signs carries no overlap
problem and the results are reliable. The approach is tested on $N_t = 4$
lattices with $2+1$ flavors and physical quark masses using the unimproved
staggered discretization. By measuring the Fisher (sometimes also called
Lee-Yang) zeros in the bare coupling on spatial lattices $L/a = 8, 10, 12$ we
conclude that the cross-over present at $\mu = 0$ becomes stronger at $\mu > 0$
and is consistent with a true phase transition at around $\mu_B/T \sim 2.4$.
