What can I do if my carb-heat cable breaks in-flight?
Well, this depends a little on your situation.  

In all cases your goal is get on the ground in a controlled manner that leaves the pilot, passengers, and airframe intact - so obviously you'll be heading for the nearest airport (and if you can get in contact with ATC and explain your situation doing so would be a really good idea).



No signs of carb ice (yet).

If you're flying in conditions conducive to carburetor icing but don't yet have signs of icing there are a few things you can do to help prevent it from forming:


Seek a warmer/drier altitude
There's a wonderful diagram I'm sure everyone has seen that gives an idea of the probability of carb ice. If you can find an altitude where the temperature/dew point spread is less conducive to carb ice formation fly there until you can land.
Lean the mixture (as practical) to limit evaporative cooling from the fuel vaporizing.
This can limit the temperature drop in the carburetor, and if it's only a few degrees of difference between ice and no ice it might help.
Avoid low-power operation (&lt;60% power).
A partially closed throttle plate is more likely to pick up ice (there is a greater pressure drop across the throttle plate which means a greater temperature drop).
"Clear" the engine periodically by opening &amp; closing the throttle.
This helps ensure the throttle plate does not freeze in place, and can dislodge any small ice patches before they become a more serious problem.



  I think this would qualify as an "urgency" situation (pan-pan), per this answer.
  If you can get in contact with ATC don't hesitate to let them know that.




Signs of carb ice are apparent.

If you do have signs of carb icing you can do all of the things above too, but how effective they are may vary.

As xpda noted adjusting the mixture can help (typically carb ice makes the engine run over-rich).
In extreme cases wide-open throttle and an aggressively leaned mixture may keep the engine producing power (and will reduce the pressure drop across the throttle plate, possibly bringing the carb temperature up enough that the ice will melt off).

In many light singles your alternate air is the carb heat box, but if you have an alternate air source for the engine besides carb heat that may be worth trying as well (especially if it's in the engine compartment or heated cabin). 


  Carburetor icing with carb heat inoperative is a bona-fide emergency.
  I can't imagine not making a mayday call in this case. Best-case scenario you wind up making an uneventful landing with fire trucks waiting to greet you ; Worst-case scenario you wind up landing somewhere that's not an airport, but at least someone knows where you are.