We present results on the QCD phase diagram for μB⩽ πT. Our simulations are performed with an imaginary chemical potential μ= iμI for which the fermion determinant is positive. On an 8 3×4 lattice with 2 flavors of staggered quarks, we map out the phase diagram and identify the (pseudo-) critical temperature Tc( μI). For μI/ T⩽ π/3, this is an analytic function, whose Taylor expansion is found to converge rapidly, with truncation errors far smaller than statistical ones. The truncated series may then be continued to real μ, yielding the corresponding phase diagram for μB≲500 MeV. This approach provides control over systematics and avoids reweighting. We compare it with other recent work.