First, You may or may not have to replace the belt.
Check to see that the idler pulley is free and thet the idler arm is also moving freely. If the idler arm is stuck, then it won't put thte correct amount of pressure on the belt. Also check the idler arm spring to make sure it is there and working.
If the belt is wet, it may slip. Dry the belt and then try it. If it still doesn't work right, you will have to replace the belt. Make sure that you are using the exact belt called for. Many people try to get a replacement belt form a hardware store or auto store that is similar and it does not work.
You really cannot see if the belt is worn in most cases. The belt wears on the sides and usually wears fairly evenly. This causes the belt to be narrower than it is supposed to be. The narrow belt, not only is not wide enough to catch the sides of the pulleys correctly, but it also falls deeper into the groves of the pulleys....thus causing the belt to appear to bee too long and not have the correct tension.
Check the entire belt system to make sure that there is no grease or oil on the bet or the pulleys. Sometimes some oil, or fuel, or greas can get onto the system...causing it to slip.
I just remembered that some of these old ATLAS snowthrowers had a special clutch on the engine that goes bad quite often. Check the end of the crankshaft on your snowthrower and see if you have a clutch on it. If this is the case, then you may have hit a dead end. These clutches have benn obsolete for many years.
If these things don't help, you may be forced to have a good shop take a look at it and find out what is wrong.