The breakout reaction 15O(alpha,gamma)19Ne, which regulates the flow between the hot CNO cycle and the rp process, is critical for the explanation of the burst amplitude and periodicity of x-ray bursters. We report on the first successful measurement of the critical alpha-decay branching ratios of relevant states in 19Ne populated via 19F(3He,t)19Ne. Based on the experimental results and our previous lifetime measurements of these states, we derive the first experimental rate of 15O(alpha,gamma)19Ne. The impact of our experimental results on the burst pattern and periodicity for a range of accretion rates is analyzed.