Relation between tensor and hom in a specific case.

Question

Suppose I know that $\mbox{Hom}_{\mathbb{Z}}(G,\mathbb{R})=0$ for a finitely generated abelian group $G$, can I then conclude that $G \bigotimes_{\mathbb{Z}} \mathbb{R}=0$ too? I know that in general Hom and 'tensoring' aren't quite the same, but are the cardinalities at least equal? I came across this when working on a different problem and I would need it for my solution. If needed I can give more context. Thanks in advance.