  In this paper we present the results of a coherent narrow-band search for
continuous gravitational-wave signals from the Crab and Vela pulsars conducted
on Virgo VSR4 data. In order to take into account a possible small mismatch
between the gravitational wave frequency and two times the star rotation
frequency, inferred from measurement of the electromagnetic pulse rate, a range
of 0.02 Hz around two times the star rotational frequency has been searched for
both the pulsars. No evidence for a signal has been found and 95$\%$ confidence
level upper limits have been computed both assuming polarization parameters are
completely unknown and that they are known with some uncertainty, as derived
from X-ray observations of the pulsar wind torii. For Vela the upper limits are
comparable to the spin-down limit, computed assuming that all the observed
spin-down is due to the emission of gravitational waves. For Crab the upper
limits are about a factor of two below the spin-down limit, and represent a
significant improvement with respect to past analysis. This is the first time
the spin-down limit is significantly overcome in a narrow-band search.

Instrumentation and Methods for Astrophysics;High Energy Astrophysical Phenomena;General Relativity and Quantum Cosmology