I would like to add a little bit to the answer above. I wrote a comment earlier, but I thought it'd be helpful to flesh out the argument a little more.
We have a firm that uses two factors of production, labor $l$ and capital $k$, to produce output. Quantity of output is written $q$.
The elasticity of a function of a single variable measures the percentage response of a dependent variable to a percentage change in the independent variable.
On the other hand, the elasticity of substitution between two factor inputs measures the percentage response of the ratio of their quantities to a percentage change in the relative marginal products.
Pertaining to the above, we have that the elasticity is given by
\begin{align}
\sigma \equiv \frac{d \ln{\left( k/l \right)}}{d\ln{ \left( MPL/MPK \right)}}
\end{align}
where $MPL$ is the marginal product of labor and $MPK$ is the marginal product of capital.
The reason I am writing this out is that there is a small error in the above answer. In the equation right after "Now let's tackle your elasticity problem," $\ln{\frac{q_k}{q_l}}$ is immediately followed by an expression for $\ln{\frac{q_l}{q_k}}$ switching the numerator with the denominator.
If you correct this, you get that $\sigma = -\frac{1}{1-\rho}$, which is close but not quite correct. To get the correct answer, you follow otherwise exactly the same calculations given by the above answer to get
$$ \ln{k/l} = \frac{1}{1-\rho}\cdot \ln {\frac{q_l}{q_k}}$$
to get that $\sigma = \frac{1}{1-\rho}$, where the corrections are for reasons outlined above.