Before you get too far. Under what exact conditions did the car stop? ( driving , sitting at a light, trying to restart? ) Did all of the electrical items in the car go out? Were they dim or completely blank?

Use these definitions for clarity.

Cranking = the key is turned and an electric motor ( starter ) rotates the engine.

Running = after cranking the engine, it consumes fuel and no longer needs the starter to rotate.

How did you determine that the battery had failed?

Failed = won't accept a charge / can't pull sufficient power out of the battery to crank the engine. ( A gas tank that has a hole in it. )

Discharged = a battery in good condition that has had electricity pulled out of it and is waiting to be recharged. ( A non leaking gas tank that is empty and is waiting to be refilled. )

I'm thinking that your first problem is a symptom to the one you are having now.