The correct cable, complete with sheathe and all, for pre 66 model trans is quite expensive compared to the 66 and later. Speedometer cable and housing for 1960-65 Plymouth Valiant, 1961-62 Dodge Lancer, 1963-65 Dart and 1964-65 Barracuda
If the sheathe is still good you can buy a universal replacement cable for it. Another member dealt with this recently. @dibbons maybe? I forget
The first thing I would do is pull the speedometer end and confirm the cable is broken. Then go under the car and pull that end of the cable to verify trans and cable are in fact pre 66 type.