In 1996, Reed proved that the domination number gamma(G) of every n-vertex graph G with minimum degree at least 3 is at most 3n/8. This bound is sharp for cubic graphs if there is no restriction on connectivity. In this paper we show that gamma(G) <= 4n/11 for every n-vertex cubic connected graph G if n > 8. Note that Reed's conjecture that gamma(G) <= inverted right perpendicularn/3inverted left perpendicular for every connected cubic n-vertex graph G is not true. (C) 2007 Elsevier B.V. All rights reserved.