We demonstrate how to realize an optical clock with neutral atoms that is competitive to the currently best single ion optical clocks in accuracy and superior in stability. Using ultracold atoms in a Ca optical frequency standard, we show how to reduce the relative uncertainty to below 10(-15). We observed atom interferences for stabilization of the laser to the clock transition with a visibility of 0.36, which is 70% of the ultimate limit achievable with atoms at rest. A novel scheme was applied to detect these atom interferences with the prospect to reach the quantum projection noise limit at an exceptional low instability of 4 x 10(-17) in 1 s.