The Ne18(α,p)Na21 reaction plays a significant role in Type-I X-ray bursts. It is a major path in the breakout from the hot-CNO cycles to the synthesis of heavier elements in the αp- and rp-processes. An experiment to determine the cross section of this reaction was performed with the ANASEN active-target detector system, determining the cross section at energies between 2.5 and 4 MeV in the center-of-mass frame. The measured cross sections for reactions populating the ground state in Na21 are consistent with results obtained from the time-inverse reaction, but significantly lower than the previously published experimental data of direct measurements. The total cross sections are also compared with those derived from indirect methods and statistical-model calculations. This experiment establishes a new experimental data set on the excitation function of the Ne18(α,p)Na21 reaction, revealing the significance of the excited states' contributions to the total reaction cross section and allowing us to separate the contribution of the (α,2p) reaction. The impact of the measured cross section on thermal reaction rates is discussed.