  We derive an explicit expression for the $1/c$ contribution to the Virasoro
blocks in 2D CFT in the limit of large $c$ with fixed values of the operators'
dimensions. We follow the direct approach of orthonormalising, at order $1/c$,
the space of the Virasoro descendants to obtain the blocks as a series
expansion around $z=0$. For generic conformal weights this expansion can be
summed in terms of hypergeometric functions and their first derivatives with
respect to one parameter. For integer conformal weights we provide an
equivalent expression written in terms of a finite sum of undifferentiated
hypergeometric functions. These results make the monodromies of the blocks
around $z=1$ manifest.
