Yes, in most cases – though you may have to call the new card provider if it can't process the transfer request online. American Express usually has 15-digit card numbers, rather than the standard 16, so some systems are unable to process the shorter number (for example, Santander, Tesco Bank and TSB have told us this is the case for most requests).
If this happens, calling the provider is your best bet or you could try adding a '0' to the beginning or end of the card number to 'trick' the system.
There are some notable exceptions though, as cards issued by NewDay such as Aqua, Fluid and Marbles above won't accept any transfer from an Amex. Equally, Sainsbury's Bank advises to call to check the Amex card(s) you hold can be accepted in advance and Virgin Money will only accept UK-issued Amex cards.