In this paper, we address the problem of generating relevant rare association rules. In the literature, this problem has not yet been studied in detail, although rare association rules can also contain important information just as frequent association rules do. Our work is motivated by the long-standing open question of devising an efficient algorithm for finding rules with low support and very high confidence. In order to find such rules using conventional frequent itemset mining algorithms like Apriori, the minimum support must be set very low, which drastically increases the runtime of the algorithm. Moreover, when minimum support is set very low, Apriori produces a huge number of frequent itemsets. This is also known as the "rare item problem". For this long-existing problem we propose a solution. A particularly relevant field for rare itemsets and rare association rules is medical diagnosis. For example it may be that in a large group of patients diagnosed with the same sickness, a few patients exhibit unusual symptoms. It is important for the doctor to take this fact into consideration.