Using PowerShell to write a file in UTF-8 without the BOM

Out-File seems to force the BOM when using UTF-8:

$MyFile = Get-Content $MyPath$MyFile | Out-File -Encoding "UTF8" $MyPath

How can I write a file in UTF-8 with no BOM using PowerShell?

Update 2021

PowerShell has changed a bit since I wrote this question 10 years ago. Check multiple answers below, they have a lot of good information!