1E 161348-5055 (1E) is a compact object lying at the center of the 2000 year old Supernova Remnant (SNR) RCW103. Its original identification as an isolated, radio-quiet neutron star has been questioned in recent years by the observation of a significant long-term variability, as well as by reports of a possible periodicity at ∼6 hours. Here we report conclusive evidence for a strong (nearly 50%) periodic modulation of 1E at 6.67±0.03 hours, discovered during a long (90 ks) XMM-Newton observation performed in August 2005, when the source was in a “low state”. The source spectrum varies along the 6.67 hr cycle. No fast pulsations are seen. 1E could be a very young binary system, possibly composed of a compact object and a low-mass star in an eccentric orbit. This would be the first example of a low-mass X-ray binary (LMXB) associated with a SNR, and thus the first LMXB for which we know the precise birth date, just 2000 years ago. Alternatively, if it is an isolated neutron star, the unprecedented combination of age, period and variability may only fit in a very unusual scenario, featuring a peculiar magnetar, dramatically slowed-down over 2000 years, possibly by a supernova debris disc.