Change User Primary Email Address In Office 365 With PowerShell without changing the username

For changing the username as well follow this link
Change a user name and email address in Office 365 with PowerShell

Open Power Shell as an Administrator

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Go to Start
Search for Power Shell
Right click on Windows Power Shell
Run as Administrator

Run the following commands

Set-ExecutionPolicy RemoteSigned

(Need to configure this setting only once on your computer)
-Press “A” or “Y”
–Press Enter

$UserCredential = Get-Credential

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Windows Power Shell Credential Request
Enter Admin’s Username & Password

$Session = New-PSSession -ConfigurationName Microsoft.Exchange -ConnectionUri https://outlook.office365.com/powershell-liveid/ -Credential $UserCredential -Authentication Basic -AllowRedirection

Import-PSSession $Session -DisableNameChecking

Following command sets a user’s primary email address from the current address to the desired address without changing the username

Set-Mailbox -identity user@current.com -WindowsEmailAddress user@new.com