All B body rearends, 62 and later, will need the spring perch center to center distance corrected to 43", but if that '62 Dart is a six cylinder and not a station wagon, it could have an 8 1/4" rearend. Look for casting #1828448 on the center section and an axle housing width of 53 1/4" flange to flange. From 57-62 this center section (8 1/4") calls for a housing with a different # (1045744 or 2070269) than the 8 3/4" rearends indicating that unlike the 63-64 8 1/4" center sections, it may have a different mounting flange reducing interchange possibilities with 8 3/4" center sections/housings. The fact that the factory considered this 8 1/4" rearend unsuitable for even a six powered station wagon should be a heads up.
Even if the '62 turns out to be an 8 3/4" rear, the small pinion and unflanged axles might mean upgrading the center section and buying aftermarket axles made to length for the housing if you plan on extreme duty and/or want the convenience of flanged axles.
Free or not, the early rearend might end up being the most expensive.
With the Duster/Demon body having the most generous rear wheelhouses of all the A bodies, while a 71 or later (over 57" housing width) might be too wide, it sounds like a later B body axle (apx. 54 1/4" to 54 7/8" housing width depending on year 65 to 70) is the way to go and even preferable when using large diameter late model wheels having deep backspacing. This would be especially true if said axle comes with one of the stronger 742 (1 3/4" pinion) or 489 (1 7/8" pinion) center sections and/or Sure Grip.