Sympathetic flares are a pair of flares that occur almost simultaneously in different active regions, not by chance, but because of some physical connection. In this paper statistical evidence for the existence of sympathetic flares is presented. From GOES X-ray flare data, we have collected 48 pairs of near simultaneous flares whose positional information and Yohkoh soft X-ray telescope images are available. To select the active regions that probably have sympathetic flares, we have estimated the ratio R of actual flaring overlap time to random-coincidence overlap time for 38 active region pairs. We have then compared the waiting-time distributions for the two different groups of active region pairs (R > 1 and R < 1) with corresponding nonstationary Poisson distributions. As a result, we find a remarkable overabundance of short waiting times for the group with R > 1. This is the first time such strong statistical evidence has been found for the existence of sympathetic flares. To examine the role of interconnecting coronal loops, we have also conducted the same analysis for two subgroups of the R > 1 group: one with interconnecting X-ray loops and the other without. We do not find any statistical evidence that the subgroup with interconnecting coronal loops is more likely to produce sympathetic flares than the subgroup without. For the subgroup with loops, we find that sympathetic flares favor active region pairs with transequatorial loops.