I've had the car for over 9 years now, so the battery is kinda old. I could not find a receipt when the battery was acquired by the previous owner. So, the battery is even older. 9+ years old battery and 6000 miles added, it still starts the first time, so the battery seem okay.
I checked the alternator with a multimeter. With the engine switched off, the battery is emitting 12.6v, which is fine. With the engine running, the battery is showing 14.6v, so the alternator is charging normally. But after about 15 minutes, the warning light starts to flicker very slightly, however, it disappears when the engine is revved. At this stage (with the warning light slightly flickering), I checked the battery voltage. It dropped to just 12.9v. It remained 12.9v when the engine is revved. I am sure, if I switched on the lights etc, it would have fallen below the 12.6v threshold of acceptability. So I think the alternator needs repairing quite soon.
I drove it for about an hour. Strangely, the flickering light disappeared! I think the alternator's charging capability is starting to deteriorate.
I also checked the belt tension and the connection behind the alternator as per Lad's suggestion. They seem fine.