What is the fabric outer bag of a hot-air balloon called?

The fabric outer bag of a hot-air balloon is called the envelope.

The envelope is the large, usually colorful structure that contains the heated air. It is commonly made from lightweight nylon or polyester fabric, with stronger materials used around the mouth, load tapes, and attachment points. Its shape and volume determine how much heated air the balloon can hold.

At the bottom, the envelope has a wide opening known as the mouth or throat. Burners mounted above the basket direct flames into this opening. At the top, many modern balloons have a parachute valve, which allows the pilot to release hot air for controlled descent and landing.

The envelope is not an airtight pressure vessel like an airplane cabin. Air flows in and out through the bottom, while the fabric mainly contains a large volume of warm air. The basket, burners, fuel tanks, suspension cables, and envelope together form the complete balloon system.