Conventional sparse phase retrieval schemes can recover sparse signals from the magnitude of linear measurements but only up to a global phase ambiguity. This work proposes a novel approach to achieve ambiguity-free signal reconstruction using the magnitude of affine measurements, where an additional bias term is used as reference for phase recovery. The proposed scheme consists of two stages, i.e., a support identification stage followed by a signal recovery stage in which the nonzero signal entries are resolved. In the noise-free case, perfect support identification is guaranteed using a simple counting rule subject to a mild condition on the signal sparsity, and the exact recovery of the nonzero signal entries can be obtained in closed-form. The proposed scheme is then extended to the sparse noise (or outliers) scenario. Perfect support identification is still ensured in this case under mild conditions on the support size of the sparse outliers. With perfect support estimation, exact signal recovery from noisy measurements can be achieved using a simple majority rule. Computer simulations using both synthetic and real-world data sets are provided to demonstrate the effectiveness of the proposed scheme.