Quantitative mapping of the longitudinal relaxation rate (R1=1/T1) in the human brain enables the investigation of tissue microstructure and macroscopic morphology which are becoming increasingly important for clinical and neuroimaging applications. R1 maps are now commonly estimated from two fast high-resolution 3D FLASH acquisitions with variable excitation flip angles, because this approach is fast and does not rely on special acquisition techniques. However, these R1 maps need to be corrected for bias due to RF transmit field (B1(+)) inhomogeneities, requiring additional B1(+) mapping which is usually time consuming and difficult to implement. We propose a technique that simultaneously estimates the B1(+) inhomogeneities and R1 values from the uncorrected R1 maps in the human brain without need for B1(+) mapping. It employs a probabilistic framework for unified segmentation based correction of R1 maps for B1(+) inhomogeneities (UNICORT). The framework incorporates a physically informed generative model of smooth B1(+) inhomogeneities and their multiplicative effect on R1 estimates. Extensive cross-validation with the established standard using measured B1(+) maps shows that UNICORT yields accurate B1(+) and R1 maps with a mean deviation from the standard of less than 4.3% and 5%, respectively. The results of different groups of subjects with a wide age range and different levels of atypical brain anatomy further suggest that the method is robust and generalizes well to wider populations. UNICORT is easy to apply, as it is computationally efficient and its basic framework is implemented as part of the tissue segmentation in SPM8.