It depends on the date.
It's not easy to tell the 1982s apart, so to figure out how many cents weigh a pound (or any other amount) you'd have to decide which date range you wanted to use and eliminate all 1982 coins. A pound is 453.6 gm so just divide that by the weight of the penny date that you decided to use.
See Related Links, below, for a list of all current specifications for other U.S. coins
Read more, below about the various pennies.
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A US current penny weighs 2.5 grams.
1000 pennies would weigh approximately 2.5 pounds.
A 1981 penny weighs 2.5 grams.
A 2012 penny weighs about 2.5 grams.
3.11 grams for pennies made from 1959 to 1982