Summary: A \(d\)-dimensional polycube of size \(n\) is a connected set of \(n\) cubes in \(d\) dimensions, where connectivity is through \((d-1)\)-dimensional faces. Enumeration of polycubes, and, in particular, specific types of polycubes, as well as computing the asymptotic growth rate of polycubes, is a popular problem in combinatorics and discrete geometry. This is also an important tool in statistical physics for computations and analysis of percolation processes and collapse of branched polymers. A polycube is said to be \(proper\) in \(d\) dimensions if the convex hull of the centers of its cubes is \(d\)-dimensional. In this paper we prove that the number of polycubes of size \(n\) that are proper in \(n-3\) dimensions is \(2^{n-6} n^{n-7} (n-3) (12n^{5} - 104n^{4} + 360n^{3} - 679n^{2} + 1122n - 1560)/3\).