  Context. Low-mass extrasolar planets are presently being discovered at an
increased pace by radial velocity and transit surveys, opening a new window on
planetary systems. Aims. We are conducting a high-precision radial velocity
survey with the HARPS spectrograph which aims at characterizing the population
of ice giants and super-Earths around nearby solar-type stars. This will lead
to a better understanding of their formation and evolution, and yield a global
picture of planetary systems from gas giants down to telluric planets. Methods.
Progress has been possible in this field thanks in particular to the sub-m/s
radial velocity precision achieved by HARPS. We present here new high-quality
measurements from this instrument. Results. We report the discovery of a
planetary system comprising at least five Neptune-like planets with minimum
masses ranging from 12 to 25 M_Earth, orbiting the solar-type star HD 10180 at
separations between 0.06 and 1.4 AU. A sixth radial velocity signal is present
at a longer period, probably due to a 65-M_Earth object. Moreover, another body
with a minimum mass as low as 1.4 M_Earth may be present at 0.02 AU from the
star. This is the most populated exoplanetary system known to date. The planets
are in a dense but still well-separated configuration, with significant secular
interactions. Some of the orbital period ratios are fairly close to integer or
half-integer values, but the system does not exhibit any mean-motion
resonances. General relativity effects and tidal dissipation play an important
role to stabilize the innermost planet and the system as a whole. Numerical
integrations show long-term dynamical stability provided true masses are within
a factor ~3 from minimum masses. We further note that several low-mass
planetary systems exhibit a rather "packed" orbital architecture with little or
no space left for additional planets. (Abridged)
