Summary: Given a graph $G = (V,E)$ and a positive integer $k$, the Proper Interval Completion problem asks whether there exists a set $F$ of at most $k$ pairs of $(V \times V) \setminus E$ such that the graph $H = (V, E \cup F)$ is a proper interval graph. The Proper Interval Completion problem finds applications in molecular biology and genomic research [11]. First announced by Kaplan, Tarjan and Shamir in FOCS ’94, this problem is known to be FPT [11], but no polynomial kernel was known to exist. We settle this question by proving that Proper Interval Completion admits a kernel with $O(k ^{5})$ vertices. Moreover, we prove that a related problem, the so-called Bipartite Chain Deletion problem admits a kernel with $O(k ^{2})$ vertices, completing a previous result of Guo [10].