Remington has a 2 or 3 letter date code. See if you can find a combination of letters somewhere near the serial number on the barrel. The first letter is the month and is one of these letters B L A C K P O W D E R X (clever eh?) B is Jan, L is Feb, etc. The year is one of these:
M - N - P - R - S - T - U - W - X - Y - Z - A(1932) - B - C - D - E - F - G - H - J - K - L - MM - NN - PP - RR - SS - TT - UU - WW - XX - YY - ZZ(1953) - A - B - C - D - E - F - G - H - J - K - L - M - N - P - R - S - T - U - W
Where the first M is 1921 and the final W is 1972.
If your rifle doesn't fit in those dates, post here and I'll see if I can find my set of codes for later years.
[Added]
Found it in the Blue Book. The series above continues with X Y Z (1975) then 1976 is I, 1977 is O, 1978 is Q, 1979 is V
Then the series is regular with 1980 = A
B C D E F G H I J K L M N O P Q R S T U W X Y Z (2004)
As an aside, I wonder what significance the IOQV has for 1976 to 1979?