How to redirect the output of a PowerShell to a file during its execution

I have a PowerShell script for which I would like to redirect the output to a file. The problem is that I cannot change the way this script is called. So I cannot do:

 .\MyScript.ps1 > output.txt

How do I redirect the output of a PowerShell script during its execution?

Maybe Start-Transcript would work for you. First stop it if it's already running, then start it, and stop it when done.

$ErrorActionPreference="SilentlyContinue"
Stop-Transcript | out-null
$ErrorActionPreference = "Continue"
Start-Transcript -path C:\output.txt -append
# Do some stuff
Stop-Transcript

You can also have this running while working on stuff and have it saving your command line sessions for later reference.

If you want to completely suppress the error when attempting to stop a transcript that is not transcribing, you could do this:

$ErrorActionPreference="SilentlyContinue"
Stop-Transcript | out-null
$ErrorActionPreference = "Continue" # or "Stop"

Microsoft has announced on Powershell's Connections web site (2012-02-15 at 4:40 PM) that in version 3.0 they have extended the redirection as a solution to this problem.

In PowerShell 3.0, we've extended output redirection to include the following streams: 
 Pipeline (1) 
 Error    (2) 
 Warning  (3) 
 Verbose  (4) 
 Debug    (5)
 All      (*)

We still use the same operators
 >    Redirect to a file and replace contents
 >>   Redirect to a file and append to existing content
 >&1  Merge with pipeline output

See the "about_Redirection" help article for details and examples.

help about_Redirection

Use:

Write "Stuff to write" | Out-File Outputfile.txt -Append

I take it you can modify MyScript.ps1. Then try to change it like so:

$(
    Here is your current script
) *>&1 > output.txt

I just tried this with PowerShell 3. You can use all the redirect options as in Nathan Hartley's answer.