The norm of an integer partition is defined as the product of its parts. This statistic was recently introduced by Schneider in connection to partition zeta functions. In this note, we use the method of moments to study the distribution of the norm under the uniform probability measure on partitions of $n$ as $n \to \infty$. We use singularity analysis to prove asymptotics for the moments and show as a result that the norm lacks a non-trivial limiting distribution on $[0,\infty)$.