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I forgot to mention one more complicating factor: the website is being viewed in full screen mode, and by going directly to a URL I do not want the URL bar to be shown.
For example, the following applescript does the job, but as a by-product the URL bar is show and does not go away again until the mouse is moved.
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tell application "Firefox" to activate
tell application "System Events"
keystroke "l" using {command down}
keystroke "My_URL" & return
end tell
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Basically I'm trying to create a headless web browser.