I feel that I must interject a couple of things. No offense, Vesi... you are substantially correct. However, the .308 and 7.62 x 51 NATO rounds are not, in fact, identical. There ARE very slight differences. Military rifle chambers are, in almost all cases, somewhat "looser" (machined to a bit looser dimensions) than those found on civilian weapons (which were designed expressly for .308 Win.). In some cases, this will cause headspace issues, when commercial ammo (.308) is used in military rifles. The looser dimensions are for the purpose of ensuring reliable feeding and extraction during periods of rapid (or full automatic) fire - at the expense of ultimate accuracy. This MAY not apply in the case of this particular rifle - IF it is a commercial version (which it very likely is) - it may be chambered for .308 anyway. There is another possibility. Generally, military brass cases (7.62 NATO) have a bit greater wall thickness than commercial brass. So, IF the rifle in question has a "military" chamber..... and/ or doesn't headspace quite right with .308... it is a bit more likely that a cartridge case rupture could occur. M1A rifles HAVE "blown up" as a result. This is not a likely occurance (it is rare) - but I think it not a good idea to take chances.
The solution(s) for these issues are not too difficult. First, check the barrel stampings. If the barrel reads ".308 Win." ... then you are probably fine (the rifle probably has a barrel machined to commercial tolerances). This is likely the case. If the barrel is NOT stamped ".308 Win.".... or if it is stamped 7.62 x 51... then, I would HIGHLY recommend that you take the weapon to a good gunsmith, to have the headspace and chamber issues checked. If there are any questions beyond this, I would recommend contacting Springfield Armory (give them the serial number) - and ASK.
I'm sure that they would be glad to assist you.
The general rule-of-thumb for this issue is: if the rifle is a COMMERCIAL (sporting) rifle type (for example, I have two Remington 700's in .308).... then it is generally fine to shoot 7.62 x 51.
I shoot 7.62 NATO all the time... with NO problems whatsoever.
If the rifle is a MILITARY type, then it is best to stay with MILITARY ammo (unless one performs the above-mentioned checks - and determines that there are no problems).
Finally, it is often true that commercial ammo is loaded to higher pressures than military fodder (again, for reasons of reliability under hard use). IF the rifle in question DOES exhibit the problems mentioned above - this fact only makes catastrophic failure more likely.
Better safe than sorry, I think. Best of Luck with this.