I had someone stall for 12 minutes before deciding to move and I am not aware or any time limit on moves. The only way I can think of to avoid this conduct is reporting the player--I do not know how to do this but the consequences could lead to the player's suspension from live chess, etc.
Stalling by opponent

@OP: Any player who behaves as you have described will be punished according to the Fair Play Policy. You don't have to report anyone who makes you wait; it is taken care of for you.

How you know that opponent was not using time to deeply think? Can you please explain that so we can get better idea of what happen?
Most of us are able to use common sense when deciding if someone is stalling or not! The fact the opponent had just lost his queen and rook in successive moves, might have had something to do with it. Also to do with the previous speed of moves made might suggest how fast the opponent might move.
To the OP, there aren't any time limits on the moves, apart from the ones for all moves agreed at the start of the game.
However I agree this is bad manners. It is the same as in an OTB game your opponent leaving the table, and simply not playing whilst your waiting for a win, whilst your opponents time runs down.
Nine or ten minutes?
Make a cup of tea; read a couple more pages of a la recherche du temps perdue; write a letter; contemplate the infinite.
What is the problem?

The origin post say nothing about opponent losing queen and rook. Nor say nothing about opponent moving quick before. Where you get this information? Make it up?
Er no I've applied my common sense from the bits of information given in the opening post.
The opening post gives the (in)complete website link to the game, it didn't take too long to search through his game history. The game was found easily enough. I found the game and played through the game from the start (something I always check before commenting on this type of forum post.)
The game was a 30 0 game meaning 30 minutes for each side to complete all their moves. The opening post says that he had to wait 20 minutes for the win, which means there could have only been about 10 minutes gone of the opponents time, in which time he has already made 19 moves. There has roughly been a move therefore every 30 seconds.
(Of course I take the OPs word that they had to wait 20 minutes at face value I have no reason not to believe that.)
Here is the complete game link: http://www.chess.com/livechess/game?id=383048031
Honestly if someone made moves every 30 seconds against me in an OTB game, and then got up from the table with 20 minutes left and just left me sitting there I would complain to the league about the poor sportsmanship. To the OP there isn't really much you can do about it here, just block him, and don't play him again.
If you look at the position there is nothing really to think about.

Honestly if someone made moves every 30 seconds against me in an OTB game, and then got up from the table with 20 minutes left and just left me sitting there I would complain to the league about the poor sportsmanship. To the OP there isn't really much you can do about it here, just block him, and don't play him again.
If you look at the position there is nothing really to think about.
And the league would say "players can think for as long as they like about a single move. Get used to it."
As for the last sentence, who are we to second-guess another game that is in progress? Imagine you started a chess game and a couple of 2800s came over and just adjudicated the game as a draw, with the words: "It's obviously a draw, there is no need to play on" and packed up the set and clock. How would you feel?
All this interrupting games is NOT a part of REAL chess.
Would the league say that if I said my opponent had got up from the table, started their car, and had gone home though.

Consider this!
A guy made 46 moves in 3 minutes or so, averaging 2 moves per second.
BUT it took him the whole 7 minutes remaining of his time to think how to avoid a checkmate!
http://www.chess.com/forum/view/general/how-to-avoid-checkmate
Useful replies, everybody. Thanks for the thought-provoking discussion. As I'm new to the site I wasn't entirely sure of all the rules. Fine, I'll wait out the time in future, should similar happen. Or catch up on my Proust, as someone suggested...
In a real-life game one at least sees the other fellow cogitating opposite you. Hence my neophyte frustration.

the opponent should just quit this is poor sportsmanship it is in effect like the kid on the playground who is losing and decides to quit and take his ball home. just as you are left with no ball.. you are also left with no opponent- i have quit many games where my opponent was taking too long for my liking. and i have also had nasty chat msg telling me to 'play' while i was thinking. but i just waited approx 15 min for my opponent to move.. which he didnt. there were 13 moves in a 30 min game. no need to debate on the intentions on this one.

...i will also add i even offered my opponent a DRAW thinking that if he didnt want to play i would just as well bag it because no one had a clear advantage. but he made a move

The guy who started this thread just stalled on me for 20 minutes when I had him clearly beat. A hypocrite at best. Try having some dignity. Judging by your low rank I expect you would be used to losing by now.
This problem is the most annoying part of chess.com. Other websites are better in this aspect.
@ trolls: this always happens right after someone blunders a queen or realizes there is an obvious mate in 1 or a pawn is unstoppable. and in real life, this is not a problem because any community deals with that kind of people easily. because the trolls dont help them.
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In the game I am playing at the moment, which has just ended, (http://live.chess.com/live?v=20121017#g=383048031) I made my last move at about 9 or 10 minutes, and my opponent has made no other move and has not responded to polite hints in the messaging box. I don't want to resign because I will lose points - on the other hand it is very annoying to have to wait 20 minutes for the game to time out. Is there any time limit on moves? Or is there any other way of avoiding this kind of conduct?
Thank you for any advice.