Turn MST of G to MST of G with one new edge
Please see my answer to the question "Updating an MST $T$ when the weight of an edge not in $T$ is decreased". 

Before we add $e'$ with weight $w(e')$, let us add $e'$ with weight so large (or, for example, larger than the weight of every other existing edge) that $e'$ will not be in any $MST$. In particular, $T$ remains an MST with this addition.

Now let us change the weight of $e'$  to $w(e')$. Then this question becomes the same question in the linked answer above.