  A weak $c$-colouring of a balanced incomplete block design (BIBD) is a
colouring of the points of the design with $c$ colours in such a way that no
block of the design has all of its vertices receive the same colour. A BIBD is
said to be weakly $c$-chromatic if $c$ is the smallest number of colours with
which the design can be weakly coloured. In this paper we show that for all $c
\geq 2$ and $k \geq 3$ with $(c,k) \neq (2,3)$, the obvious necessary
conditions for the existence of a $(v,k,\lambda)$-BIBD are asymptotically
sufficient for the existence of a weakly $c$-chromatic $(v,k,\lambda)$-BIBD.
