This paper contains two results on the asymptotic behavior of uniform probability measure on partitions of a finite set as its cardinality tends to infinity. The first one states that there exists a normalization of the corresponding Young diagrams such that the induced measure has a weak limit. This limit is shown to be a $\delta$-measure supported by the unit square (Theorem 1). It implies that the majority of partition blocks have approximately the same length. Theorem 2 clarifies the limit distribution of these blocks. The techniques used can also be useful for deriving a range of analogous results. Bibliography: 13 titles.