We report the discovery of the likely IR counterpart to the Anomalous X-ray Pulsar (AXP) 1RXS J1708-4009, based on the combination of Chandra HRC-I X-ray position, and deep optical/IR observations carried out from ESO telescopes and the Canada France Hawaii Telescope (CFHT) during 1999-2002. Within the narrow uncertainty region we found two relatively faint (K'=20.0 and K'=17.53) IR objects. Based on their color and position in the J-K' versus J-H diagram only the brighter object is consistent with the known IR properties of the counterparts to other AXPs. No variability was detected for this source, similarly to what is observed in the case of 4U 0142+614. Like in other AXPs, we found that the IR flux of 1RXSJ1708-4009 is higher then expected for a simple blackbody component extrapolated from the X-ray data. If confirmed, this object would be the fourth IR counterpart to a source of the AXP class, and would make the IR excess a likely new characteristic of AXPs.