  (abridged) Recent investigations aimed at estimating the distance to the
young LMC cluster NGC1866 have made use of Red Clump stars in the surrounding
LMC field, together with empirical and theoretical Main-Sequence fitting
methods, and have found significantly different distances for the field and the
cluster, the latter being closer by in distance modulus by \Delta (DM) \sim
0.20 mag. In this paper we (re-)consider the Cepheid star population of
NGC1866, to try to shed some light on this discrepancy. By combining various
extensive photometric datasets in BVI and single-epoch 2MASS JHK photometry, PL
relationships for the cluster Cepheids are obtained. A comparison between the
field LMC and cluster PL relationships for the reddening free Wesenheit index
gives a firm determination of the distance between the cluster and the LMC main
body (0.04 mag in distance modulus, the cluster being more distant) which,
coupled to a model for the geometry of the LMC disk, provides \Delta (DM)
ranging between 0.0 and -0.11 mag. The simultaneous comparison of the PL
relationships in BVI for the cluster and LMC field gives an estimate of the
cluster reddening, which results to be E(B-V) = 0.12 \pm 0.02. This
determination is higher than the canonical value of 0.06 mag used in all
previous studies, but we show that it is not in contradiction with a
re-analysis of independent estimates. The cluster Main Sequence fitting
distance obtained with this new reddening is DM = 18.58 \pm 0.08, fully
compatible with the Red Clump value of DM = 18.53 \pm
0.07(random)^{+0.02}_{-0.05} (systematic) and the Cepheid constraint on \Delta
(DM).
