After some frantic googling I do believe the answer is yes, but more so I am frustrated that the relation between the two parameters seems to be nowhere described explicitely so I do it here. (I hope this isn't against the rules of stackexchange.)
This very nice article states: we will denote the random variable Y having a negative binomial distribution as Y ~ NB($\mu, \kappa$) with a parameterization such that E(Y) = $\mu$, var(Y) = $\mu + \kappa \mu^2$.
I take this latter equation as the definition of $\kappa$.
Apparently this kappa is implemented in SAS.
Now turning to R, the function
glm.nb in the
MASS package contains a parameter $\mu$ which is obviously the same $\mu$ as above and a parameter $\theta$. The question is how $\theta$ and $\kappa$ are related. The documentation for
glm.nb only refers to it as an "additional parameter". The answers to this and this stackexchange questions directly imply that $\theta = 1/\kappa$, but this question [EDIT: since removed] seems to suggest that $\theta = \kappa$.
The help page for negative binomial in R is nice and introduces a parameter called
size that equals $1/\kappa$. Fitting
glm.nb on random data generated by
rnbinom for various choices of $\mu$ and
size seems to support the thesis that $\theta = 1/\kappa$ (i.e. that $\theta$ =
size) but also that for large values of size the estimation is poor.
Summarizing: I do believe that $\theta = 1/\kappa$ but it would be nice if there were an easily googlable place on the internet stating this explicitly. Maybe one of the answers to this questions can serve as such a place?