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All approaches currently used to study finite baryon density lattice QCDsuffer from uncontrolled systematic uncertainties in addition to the well-knownsign problem. We formulate and test an algorithm, sign reweighting, that worksdirectly at finite $\mu = \mu_B/3$ and is yet free from any such uncontrolledsystematics. With this algorithm the {\em only} problem is the sign problemitself. This approach involves the generation of configurations with thepositive fermionic weight $|{\rm Re\; det} D(\mu)|$ where $D(\mu)$ is the Diracmatrix and the signs ${\rm sign} \; ( {\rm Re\; det} D(\mu) ) = \pm 1$ arehandled 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 tothe positive weight) this discrete reweighting by the signs carries no overlapproblem and the results are reliable. The approach is tested on $N_t = 4$lattices with $2+1$ flavors and physical quark masses using the unimprovedstaggered discretization. By measuring the Fisher (sometimes also calledLee-Yang) zeros in the bare coupling on spatial lattices $L/a = 8, 10, 12$ weconclude that the cross-over present at $\mu = 0$ becomes stronger at $\mu > 0$and is consistent with a true phase transition at around $\mu_B/T \sim 2.4$.

New approach to lattice QCD at finite density; results for the critical end point on coarse lattices