In the favoured core-accretion model of formation of planetary systems, solid planetesimals accumulate to build up planetary cores, which then accrete nebular gas if they are sufficiently massive. Around M-dwarf stars (the most common stars in our Galaxy), this model favours the formation of Earth-mass (M⊕) to Neptune-mass planets with orbital radii of 1 to 10astronomical units (AU), which is consistent with the small number of gas giant planets known to orbit M-dwarf host stars. More than 170 extrasolar planets have been discovered with a wide range of masses and orbital periods, but planets of Neptune's mass or less have not hitherto been detected at separations of more than 0.15AU from normal stars. Here we report the discovery of a 5.5+5.5-2.7 M⊕ planetary companion at a separation of 2.6+1.5-0.6AU from a 0.22+0.21-0.11 Msolar M-dwarf star, where Msolar refers to a solar mass. (We propose to name it OGLE-2005-BLG-390Lb, indicating a planetary mass companion to the lens star of the microlensing event.) The mass is lower than that of GJ876d (ref. 5), although the error bars overlap. Our detection suggests that such cool, sub-Neptune-mass planets may be more common than gas giant planets, as predicted by the core accretion theory.