The gas inside will want to expand further, but it will be restrained by the envelope. Imagine the outside air pressing inward until there is an equilibrium. If the balloon then rises again (for instance, because it warms up) then it can get to a point where the outward force on the envelope exceeds the capacity of the envelop material to restrain that force.

For flexible envelopes this is more or less inevitable (balloon rises->atmospheric pressure drops->balloon expands->density decreases->balloon rise etc), for more rigid envelopes it is a balance that may work out in favor of the balloon staying in one piece (oscillating in altitude as it cools down/heats up again with the day/night cycle), or it may burst depending on the pressure differential. Most of them are pretty flimsy.

From: http://www.stratostar.net/faq.html

"When a balloon is filled on the ground with lift gas (helium or hydrogen), it can range in size from 2.5 ft to 8 ft in diameter. During the balloon's flight it will grow more than 4 times the diameter and upto 83 times the volume measured at launch, until it can't strech any more and will burst! A high-altitude weather balloon filled with 268 cu/ft of helium will have a diameter of about 8 ft at sea level, but as the balloon climbs through the atmosphere it will expand to 35ft in diameter and will have a volume of 22,449 cu/ft before it pops."