  A conjecture of Morel asserts that the sheaf of $\mathbb A^1$-connected
components of a space is $\mathbb A^1$-invariant. Using purely
algebro-geometric methods, we determine the sheaf of $\mathbb A^1$-connected
components of a smooth projective surface, which is birationally ruled over a
curve of genus $>0$. As a consequence, we show that Morel's conjecture holds
for all smooth projective surfaces over an algebraically closed field of
characteristic $0$.
