Volume preserving embeddings of open subsets of $R^n$ into manifolds

We consider a connected smooth $n$-dimensional manifold $M$ endowed with a volume form $\mathrm{\Omega }$, and we show that an open subset $U$ of $R^n$ of Lebesgue measure $\text{Vol}(U)$ embeds into $M$ by a smooth volume preserving embedding whenever the volume condition $\text{Vol}(U)\le \text{Vol}(M,\mathrm{\Omega })$ is met.