The dissociative recombination (DR) of the helium dimer ³He ⁴He⁺ has been investigated at the heavy-ion Test Storage Ring (TSR) in Heidelberg at relative energies up to 40 eV. The vibrational level population in the stored ion beam was shown to be non-thermal with a fraction of 0.1–1% in higher vibrational states, resulting mainly from vibrational excitation in collisions with the residual gas species. The temporal evolution of the DR rate during storage showed evidence for an electron induced rotational de-excitation from the vibrational ground state. After characterizing the evolution of the rovibrational population of the stored ions, the zero energy DR rate coefficient was extracted from the measurement to be α^v=0_DR (0)=(7.3 ± 2.1) × 10⁻¹⁰ cm³/s, and the DR reaction from the vibrational ground state was seen to proceed mainly to the 1s2s ¹S and 1s2p³P atomic limits. For v ≥ 3, the DR reaction has a rate coefficient ≥ 2 × 10⁻⁷ cm³/s and leads primarily to atomic fragments with n ≥ 3. The energy dependent rate coefficient displays several prominent structures.