I had a look at the PSC site referenced above in JohnRPollard's post and found the technique I had tried to describe earlier. Looks like you don't have to move the engine.
Here's the technique followed by a PS 37 owner:
After a month of shore leave, we got back to the boat
on Jan 2 and started work. I had asked this board for
information on the cutless bearing on PS 37's and
received many great replies. When it came time to
remove the old bearing, I tried a technique that I
developed to avoid dropping the rudder and pulling the
prop shaft. It worked! It depends on having sleeves
that will slide on the shaft and inside the stern
tube. PS 37's have a 1" diameter shaft and a 1 3/8" ID
stern tube. 1", schedule 40 pipe has a 1.315" OD and a
1.037" ID, so it fit's the bill.
Remove the prop
Remove two set screws holding cutless bearing
Remove the coupling from the shaft
Remove the packing gland
Cut about thirty 1" long pieces of 1", schedule 40,
Slide PVC sleeves on prop shaft until a less than 1"
gap between last sleeve and motor half of coupling.
Use wedges between last PVC sleeve and motor half of
coupling to push sleeves against end of cutless
Like magic, the cutless bearing came out of the hole!
Roland PS 37 #347