Catalyzes the phosphorylation of ribose 1,5-bisphosphate to 5-phospho-D-ribosyl alpha-1-diphosphate (PRPP). alpha-D-ribose 1,5-bisphosphate + ATP = 5-phospho-alpha-D-ribose 1-diphosphate + ADP Metabolic intermediate biosynthesis; 5-phospho-alpha-D-ribose 1-diphosphate biosynthesis; 5-phospho-alpha-D-ribose 1-diphosphate from D-ribose 5-phosphate (route II): step 3/3. In the C-terminal section; belongs to the ribose 1,5-bisphosphokinase family.