Serving a subdomain content under another domain?

I don't know if this possible or what is the technical name for this.
Say, I have a website in the subdomain "redwidgets.example.com".

Now, I bought "redwidgets.com" domain and I want that "redwidgets.com" to serve the content on "redwidgets.example1.com", without moving the files from one place to another, nor reinstalling the website, I mean, just by DNS tricks.
Is this possible?
 
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