When calling a number in the UK, the person you're contacting may be able to see your number on her cell phone or landline display screen. You can mask your number, however, by inputting a few digits before you make a call, putting the call through the operator or requesting a permanent block.
Press "141" before you enter the full telephone number of the person you're calling. This is the UK equivalent of pressing "_67" in the U.S. to hide caller display. If you're phoning the UK from the U.S., then use "_67" to hide your ID. However, this method is single-use only. The next time you make a call from the same phone and want to hide the ID, you need to enter "141."
Dial "100" from a UK land line to connect with a UK telephone operator and request to be transferred to a UK number. It will show up as a different number at the other end. In the U.S., dial "0" to reach the operator.
Contact your telephone service and ask for a permanent block on your caller ID. Most networks should be able to comply.
Using "141" to hide your number doesn't work on all UK cell phones. Some use "#31#" instead. Check with your provider.