The consumer gadget and computer maker has now surpassed Microsoft’s record of $620.58bn set in 1999. However that figure is not adjusted for inflation.
The news comes ahead of the anticipated launch of the iPhone 5, and possibly a smaller and cheaper iPad.
Apple shares hit $664.74 in New York midday trading, before falling to $663.
That was $14.98, or 2.3%, higher than Friday’s close.
There is also speculation that Apple plans to make a TV set.
However, despite its market valuation, Apple, like many US companies, faces a number of challenges.
The strength of the US dollar against the euro and other currencies makes US-made goods more expensive overseas.Added to that, the faltering economic recovery in the United States, combined with recession in major markets such as Europe, is also making it more difficult to sell consumer electronics.
Apple also faces stiff competition from Samsung’s Galaxy S3 and HTC’s One X smartphones.