The 18O(p,α)15N reaction affects the synthesis of 15N, 18O and 19F isotopes, whose abundances can be used to probe the nucleosynthesis and mixing processes occurring deep inside asymptotic giant branch (AGB) stars. We performed a low-background direct measurement of the 18O(p,α)15N reaction cross-section at the Laboratory for Underground Nuclear Astrophysics (LUNA) from center of mass energy Ec.m. = 340 keV down to Ec.m. = 55 keV, the lowest energy measured to date corresponding to a cross-section of less than 1 picobarn/sr. The strength of a key resonance at center of mass energy Er = 90 keV was found to be a factor of 10 higher than previously reported. A multi-channel R-matrix analysis of our and other data available in the literature was performed. Over a wide temperature range, T = 0.01- 1.00 GK, our new astrophysical rate is both more accurate and precise than recent evaluations. Stronger constraints can now be placed on the physical processes controlling nucleosynthesis in AGB stars with interesting consequences on the abundance of 18O in these stars and in stardust grains, specifically on the production sites of oxygen-rich Group II grains.