As requested, here is the diagram to upgrade a 600XL to 64K.

This is excerpted from a Michigan Atari Magazine article by Don Neff(MACE,
WAUG).

This is the upgrade in its simplest form.  If you do not understand
it, don't do it.  You assume all risk, but I have used this several
times and it works just fine.

You need: 2 41464-10 64K X 4 bit Dram chips, sockets optional.
some hookup wire, soldering iron, and tools

Open the case and remove the circuit board.  With the cart port at the
back...

Replace the 2 16K drams.  They are U11 & U12.  If they are socketed,
then Great.  If not, carefully unsolder them, the fine traces are on
top!  Pull up the (3)pins and fold over the fine lead, pull up the
resistor lead.  Unsolder as needed.  You may want to tape the
resistor/jumper.  I.C. notches go left.

RAM:(Replace 2 dram chips, add jumper):

                       | | | | | | | | |
                      +-----------------+
                 U12  |)    41464       |      o   o
                      +-----------------+      |   |
                      1| | | | | | | | |       -   -
           Bend                               |R| |R|
            up         | | | | | | | | |      |3| |3|
           Pin 8      +-----------------+     |6| |8|
              +---+   |)    41464       |      -   -
  | | | | | | |8  |   +-----------------+     /    |
 +-------------+  |   1| | | | | | | | |      |o   o
 |)   74LS51   |  | U11                       |  unsolder/pullup
 +-------------+  +---------------------------+ lead
 1| | | | | | |    ADD jumper:pin 8 to resistor
                              R36


ADDRESSING(add 2 jumpers):

                                          | | | | | | | | 
                                         +---------------+
                                      U5 |)  74LS158     |
                                         +---------------+
                                         1| | | | | | | |
                                 +------------+ Pin 3, bend up
                                 |
                                 | +------------------+ Pin 10, Bend up
                                 | |      | | | | | | | |
         +-----------------------+ |     +---------------+
         | +-----------------------+     |)  74LS158     |
         | |   Add two jumpers,          +---------------+
     o o o o <-solder to pads            1| | | | | | | |
     | | | |                            U6
     / / / /
    / / / /
   / / / /
   | | | |
 ---------+
          | U9
     -----+


You are done.  Test the board, FRE(0) should be about 37K.
Reassemble.