I believe the usual path taken when converting the Cadet to .357 Magnum is to have the barrel rebored and the rifling recut. There are several companies who do this and would certainly be less expensive than rebarreling and maybe even relining. Here is one that I've read good things about but I have no personal experience. 
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Bill

FWIW,
I have a Martini Cadet that was rebored in the Mid- late 
1950s for Klein's Sporting Goods in Chicago. Other than 
the rebore it is "box stock" . 
It is now a 357Mag/16 twist.
I remember them in the gun mag ads when I was in high school. Something like $29.95. Cost me a lot more than that several yrs ago when i acquired it in a trade. 

Amongst other cast bullets, it shoots 200 gr Lyman 358315 bullets real well over a case full of IMR4227.
 
I just checked. Yes, they had bushed the firing pin
hole. Good idea!.

The rifle is a great and accurate "plinker", notwithstanding the very narrow front sight and "V" rear notch are tough on this ol' eye.
beltfed/arnie

contact Steve Durren at Johnsons Sporting Goods in Adrian Michigan. 517-265-2542.  We have other excellent gunsmiths associated with this forum community that others may recommend as well, but Steve is the one I have had work on a number of single shots and I can recommend him based on my personal experience. 
  the work can be done by installing a new barrel, relining by drilling out the old rifling and permanently inserting a pre-rifled liner and then cutting a new chamber,  or the original barrel can be re-rifled--reamed out and new rifling cut.   All 3 methods have pros and cons and Steve can give you fair and honest advice about them and what work your rifle would require. 
There are not too many re-riflers are around any more.  There is one that I am aware of only works with 35 caliber and up so he might be able to do the job for a 357 mag pistol cartridge.

Fwiw many years ago my very first single shot was a beat up Winchester lo-wall with a corroded 32-20 barrel.  I had the late barrel-smith Harry McGowan ream/re-rifle/rechamber it to .357 and it was a sheer joy to shoot.  Unfortunately I ran into some financial problems and had to sell it; much to my long enduring sorrow.