We continue our investigation of graphs G for which the least eigenvalue ?(G) is minimal among the connected graphs of prescribed order and size. We provide structural details of the bipartite graphs that arise, and study the behaviour of ?(G) as the size increases while the order remains constant. The non-bipartite graphs that arise were investigated in a previous paper [F.K. Bell, D. Cvetkovic', P. Rowlinson, S.K. Simic', Graphs for which the least eigenvalue is minimal, I, Linear Algebra Appl. (2008), doi: 10.1016/j.laa.2008.02.032]; here we distinguish the cases of bipartite and non-bipartite graphs in terms of size. Erratum is published in: Richard A Brualdi, 'From the Editor-in-Chief', Linear Algebra Applications, 432(1) pp.1-6, 01/2010