We identify and rectify a crucial source of bias in the initiator full configuration interaction quantum Monte Carlo algorithm. Noninitiator determinants (i.e., determinants whose population is below the initiator threshold) are subject to a systematic undersampling bias, which in large systems leads to a bias in the energy when an insufficient number of walkers are used. We show that the acceptance probability (pacc), that a noninitiator determinant has its spawns accepted, can be used to unbias the initiator bias, in a simple and accurate manner, by reducing the applied shift to the noninitiator proportionately to pacc. This modification preserves the property that in the large walker limit, when pacc → 1, the unbiasing procedure disappears, and the initiator approximation becomes exact. We demonstrate that this algorithm shows rapid convergence to the FCI limit with respect to the walker number and, furthermore, largely removes the dependence of the algorithm on the initiator threshold, enabling highly accurate results to be obtained even with large values of the threshold. This is exemplified in the case of butadiene/ANO-L-pVDZ and benzene/cc-pVDZ, correlating 22 and 30 electrons in 82 and 108 orbitals, respectively. In butadiene 5 × 107 and in benzene 108 walkers suffice to obtain an energy within a millihartree of the coupled cluster singles doubles triples and perturbative quadruples [CCSDT(Q)] result in Hilbert spaces of 1026 and 1035, respectively. Essentially converged results require ∼108 walkers for butadiene and ∼109 walkers for benzene and lie slightly lower than CCSDT(Q). Owing to large-scale parallelizability, these calculations can be executed in a matter of hours on a few hundred processors. The present method largely solves the initiator-bias problems that the initiator method suffered from when applied to medium-sized molecules.