Right. I've seen this before. Some handyman put in a subpanel, but only ran 120V service to it (who knows why - saving money on wires? Expanding one wire feeding an outlet into a whole sub-panel?) and then jumpers the hots - which works OK if everything is 120V loads - which was probably the case when the panel was installed.
This does, of course, mean that a 2-pole breaker has pretty much 0 volts across it, as both sides are fed from the same phase (both are 120V to neutral/&ground.)
You'll need to add another hot wire from the source to the panel (traditionally red in the US system - black hot, red hot, white neutral (grounded), green or bare safety ground.)
Depending what the current wiring is, and especially if it's inappropriately sized (which would go right along with the handyman's decision to run it 120V) it may be more appropriate to replace the current wiring completely with 4 wire cable (called 3-wire cable due to the convention of not counting the ground wire) or 4 wires in conduit.
I'd also double-check the status of the ground and neutral - in a sub-panel the neutral is supposed to be isolated from the ground (including not being connected to the case, which should be grounded) and I would not bet on that being how this one is, just following the pattern of behavior shown by whoever wired it up.
You should probably go over the whole electrical system looking for any other "interesting" work that's been done to it, and possibly call in an electrician to check you on anything you can't sort out. Incidentally (clarifying my comment), both QO and Homeline are fine, it's just that the breakers have to be the same flavor as the panel, and sometimes folks get the wrong one as both are Square-D products, and they almost-sort-of fit. But that has nothing to do with your problem, from your description in the comments.
Sounds like a dead thermostat.
When it was working before, it hadn't died yet.
Why it died is likely to remain a mystery. Things don't live forever (neither do people - dragons might, per one song.)
Replace it.
Best Answer
I'd be rather surprised if your electric baseboard is actually 120V - that's fairly uncommon. 240V is much more common, without a neutral in normal use (ground, L1 (black) L2 (white, which should be taped/re-colored red, but often isn't) and 240V between L1 and L2.
Whether the heater is actually 120V or 240V, the circuit is clearly in error if there is only 0.2V between white and black.