  Natal kicks are one of the most debated issues about double neutron star
(DNS) formation. Several observational and theoretical results suggest that
some DNSs have formed with low natal kicks ($\lesssim{}50$ km s$^{-1}$), which
might be attributed to electron-capture supernovae (ECSNe). We investigate the
impact of ECSNe on the formation of DNSs by means of population synthesis
simulations. In particular, we assume a Maxwellian velocity distribution for
the natal kick induced by ECSNe with one dimensional root-mean-square
$\sigma_{\rm ECSN} = 0,7,15,26,265$ km s$^{-1}$. The total number of DNSs
scales inversely with $\sigma_{\rm ECSN}$ and the number of DNS mergers is
higher for relatively low kicks. This effect is particularly strong if we
assume low efficiency of common-envelope ejection (described by the parameter
$\alpha=1$), while it is only mild for high efficiency of common-envelope
ejection ($\alpha{}=5$). In most simulations, more than 50 per cent of the
progenitors of merging DNSs undergo at least one ECSN and the ECSN is almost
always the first SN occurring in the binary system. Finally, we have considered
the extreme case in which all neutron stars receive a low natal kick
($\lesssim{}50$~km~s$^{-1}$). In this case, the number of DNSs increases by a
factor of ten and the percentage of merging DNSs which went through an ECSN is
significantly suppressed ($<40$ per cent).
