Mapping the longitudinal relaxation time T1 has widespread applications in clinical MRI as it promises a quantitative comparison of tissue properties across subjects and scanners. Due to the long scan times of conventional methods, however, the use of quantitative MRI in clinical routine is still very limited. In this work, an acceleration of Inversion-Recovery Look-Locker (IR-LL) T1 mapping is presented. A model-based algorithm is used to iteratively enforce an exponential relaxation model to a highly undersampled radially acquired IR-LL dataset obtained after the application of a single global inversion pulse. Using the proposed technique, a T1 map of a single slice with 1.6mm in-plane resolution and 4mm slice thickness can be reconstructed from data acquired in only 6s. A time-consuming segmented IR experiment was used as gold standard for T1 mapping in this work. In the subsequent validation study, the model-based reconstruction of a single-inversion IR-LL dataset exhibited a T1 difference of less than 2.6% compared to the segmented IR-LL reference in a phantom consisting of vials with T1 values between 200ms and 3000ms. In vivo, the T1 difference was smaller than 5.5% in WM and GM of seven healthy volunteers. Additionally, the T1 values are comparable to standard literature values. Despite the high acceleration, all model-based reconstructions were of a visual quality comparable to fully sampled references. Finally, the reproducibility of the T1 mapping method was demonstrated in repeated acquisitions. In conclusion, the presented approach represents a promising way for fast and accurate T1 mapping using radial IR-LL acquisitions without the need of any segmentation.