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Depends on how many studs you break of the heater box while trying to remove it. Drain the coolant, disconnect the hoses, remove glove box, remove heater box inside and out. Then replace heater core But while the heater box is out it’s a good time to put new seals in the heater box
Just replaced mine on a 64 convertible. It’s not hard, just time consuming, and at times very frustrating. Glove box and gauge cluster needed to come out first. I don’t believe a 63 has a stud on the firewall behind the dash like a 64 which attaches to the heater box, so you may be able to leave the gauge cluster in the car. Once I got the heater box out, I order a complete heater box rebuild kit from Zip. Everything internal was rusted and/or disintegrated. New seals for the box/firewall itself, as well as the metal retainer for the actual heater core, and (obviously) a new heater core. Heater core was not an exact match dimensionally for my leaking original, but it does fit. You can do it. Just take your time and take lots of pics. I also did it alone. Another set of (small) hands would have been helpful at times. Post questions if you run into issues. I’ll watch the thread.
If you have a comfortable garage, tunes or a game on in the background, cold beer, some hand tools and the shop manual, it’s not difficult. Just take your time and don’t force anything.
It's not a terribly difficult job. I did mine on my 66. I think I removed the battery so I could see up behind there to get the corner studs. I don't recall my gauges having to come out, but the rest of the items mentioned above did come out. Maybe it's different for you fancy 63 guys.
Mine turned into a bunch of other projects (Painting the speaker cover, new carpet padding, new heater hoses and a general clean up of the entire area.
Rewarding job if you just take care and don't try to rush the job.