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This great video on Vimeo shows how close these 2 movies link together.
Obviously firing the spoiler horn here!
The end of Rogue One, where Darth Vader is shown as the endboss he is, leads perfectly to the start of Episode IV. It is not really clear how long the chase in the meantime actually is, maybe days, weeks or months, but this is where it starts.
"Rogue One" Spliced with "A New Hope" v.2 from Barre Fong on Vimeo.