As I was reading Yemon Choi's comment above it occurred to me that the situation of the crossed product $C(S^1)\times_\theta\mathbb{Z}$ is in fact a bit peculiar since the most standard representation of $C(S^1)$ one usually thinks of, namely as multiplication operators on $L^2(S^1)$, already comes equipped with a unitary representation of $\mathbb{Z}$ implementing the action by rotation.

This is not always the case for a general crossed product $A\times\mathbb{Z}$, so one usually starts with one's favorite representation of $A$ on some Hilbert space $H$ and builds the "regular representation" of the crossed product on the Hilbert space $H\otimes \ell^2(\mathbb{Z})$.

Even though that was not the representation the OP had in mind it is interesting to observe that, if the irrational rotation C*-algebra is completed in this other representation, one does indeed get a type $II_1$ factor, partly because the standard trace is a vector state in this representation and hence duly extends to a normal state on the weak closure.

PS: It was my original intention to reply to a comment by Yemon Choi, but I could not fit all of this within the 600 charactes size limitation. I therefore hope to be excused for shamelessly attempting to sidestep the rules and I am ready to delete this post should anyone complain!