Size of 240v service panel breaker
Nope. You can't run that cable through that conduit.

8/3 "underground Romex" as you call it, actually UF-B cable, is not permitted in that conduit because the cable is too large for the conduit.  So are most cables.   You are actually better off running the type that is already there, THWN-2 individual wires.  You will need 1 white and 2 of black or a color. Your existing ground wire is fine and may be reused, if it is 10 AWG.  

Romex in conduit (which will not fit) is good for 40A.  THWN wires are allowed 50A.  

Supply breaker ampacity is decided by the cable.


10 AWG wire - 30A
8 AWG NM or UF cable - 40A 
8 AWG THWN wire in conduit - 50A
6 AWG NM or UF cable - 60A
6 AWG THWN wire in conduit - 70A


If the subpanel is main-lug, the breaker also can't be larger than the busing on the subpanel, however that won't be a problem if you listen to my "think big" advice above...

Don't scrimp on subpanel size

Install the biggest subpanel you can possibly stand to get.  Panel spaces are cheap.  Running out of spaces in a panel later is an expensive, frustrating problem.  

If you aim to scrimp, then a) skip the main breaker. Your subpanel does not need it if it's in the same building.  And b) use aluminum wire, replacing #8Cu with #6Al... or #6Cu with #4Al. Aluminum wire is actually preferred at these large sizes. The subpanel input lugs will be aluminum!