You will have to ring customer service to request a change in a phone number, but it is not free, so there will usually be a charge, and they should be able to do this without the need of a new SIM card.
To contact EE customer service dial 150 from your mobile phone or ring customer service for free using web based app or another phone: +44 800 079 8586 or +44 800 956 6000.
If you are on PAYG you could just order a new PAYG SIM. Otherwise on contract it would be chargeable ~ £36.
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To phone EE CS: Dial Freephone +44 800 079 8586 - Option 1 for Mobile Phone & Mobile Broadband or Option 2 for Home Broadband & Home Phone
ISPs: 1999: Freeserve 48K Dial-Up > 2005: Wanadoo 1 Meg BB > 2007: Orange 2 Meg BB > 2008: Orange 8 Meg LLU > 2010: Orange 16 Meg LLU > 2011: Orange 20 Meg WBC > 2014: EE 20 Meg WBC > 2020: EE 40 Meg FTTC > 2022:EE 80 Meg FTTC SoGEA > 2025 EE 150 Meg FTTP
