This work demonstrates a new variant of the 3DREAM sequence for whole-brain mapping employing a three-dimensional (3D) stack-of-spirals readout. The spiral readout reduces the echo train length after the STEAM preparation in order to overcome the significant blurring in STE* images due to the decreasing STE* signal with each excitation pulse. The 3DREAM sequence rapidly acquires two contrasts to calculate whole-brain flip angle maps. In the proposed spiral 3DREAM sequence, the Cartesian readout scheme is replaced by an accelerated 3D stack-of-spirals readout with a CAIPIRINHA sampling scheme. Phantom experiments were conducted to compare flip angle maps of the spiral 3DREAM sequence to a Cartesian 3DREAM sequence, an actual flip-angle-imaging (AFI) sequence, the dual-angle method, and the Bloch-Siegert shift method. Afterwards, the results were validated in vivo acquiring flip angle maps from five subjects. Flip angle maps of the spiral 3DREAM sequences showed high agreement with the reference methods both in phantom and in vivo experiments. Blurring in STE* images and flip angle maps was reduced compared to the Cartesian 3DREAM sequence. The spiral 3DREAM sequence utilizes a fast readout minimizing the echo train length of the imaging train. This reduces blurring in STE* images as well as the total acquisition time and increases the effective resolution of maps.