What is the value of a 1963 US 1 dollar bill?

As of 08/2013 a circulated 1963 $1 bill retails for about $1.50 regardless of its series letter (none, A, or B). If it's uncirculated, it might go for around $2.00 to $2.50 at retail.

Serial Numbers

A serial number hardly ever matters to a bill's value. The only ones that are of interest are ones that are very low; e.g. 00000002, or follow a pattern such as 12345678 or 12344321.