In the Ninth Century, Cornwall was annexed into the Kingdom of Wessex by King Egbert. This was recorded in the Anglo-Saxon Chronicle, which is one of best primary sources for early medieval England (though it does have its flaws, particularly with dating.)
Here are the two excerpts from the Chronicle that are relevant.
"A.D. 813. This year Archbishop Wulfred returned to his own see, with the blessing of Pope Leo; and King Egbert spread devastation in Cornwall from east to west."
"A.D. 835. This year came a great naval armament into West-Wales, where they were joined by the people, who commenced war against Egbert, the West-Saxon king. When he heard this, he proceeded with his army against them and fought with them at Hengeston, where he put to flight both the Welsh and the Danes."
One thing to be weary of with the Chronicle is dates are sometimes a little off. The 835 entry is talking about the Battle of Hingston Down which actually took place 3 years later in 838. In this battle the Cornish allied with the Viking against Wessex, King Egbert wins the battle and this seems to be where Cornish Independence ends and it is annexed into Wessex.
The Kingdom of Wessex would then go on to become the kingdom of England under Alfred the Great and his successors. So Cornwall lost its Independence to the kingdom which would become England. (The House of Wessex ruled England from the kingdom's foundation until the Norman Conquest, aside from two brief interludes where Danish King Sven Forkbeard, his son Cnut, and two grandsons briefly ruled England)
Sources:
The Anglo-Saxon Chronicle, which you can find online here: http://avalon.law.yale.edu/subject_menus/angsax.asp
/u/mikedash has a truly excellent post on this subject (Dumnonia specifically):