Results of 27 Al-NMR measurements on the heavy-electron compound CeAl 3 , recorded in low magnetic fields (of the order of 1kG), very low temperatures (T<1.5K), and at hydrostatic pressures up to 5.6kbar are presented. We observe drastic changes of the NMR spectra and of the spin-lattice relaxation rates T1-1 with increasing pressure. For p<1.2kbar, the 27 Al spectrum is rather broad and complicated, but the situation changes significantly at higher pressures. The spectra narrow rapidly and can be reproduced by simulation calculations using reasonable parameters. The spin-lattice relaxation rate drops by almost two orders of magnitude upon increasing the pressure up to 5.6kbar, indicating a substantial decrease of the density of electronic states at the Fermi energy. The data suggest a simple paramagnetic ground state for p>1.2kbar, without any evidence of magnetic order above 65mK.