Captain Ballance followed up his 108 in the first innings with his 32nd first class ton.
He looked completely untroubled once again, especially with the old ball proving no danger on an unresponsive pitch.
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Ballance put on 88 with Australian star Peter Handscomb as Yorkshire attempted to pile on the runs before the new ball.
Handscomb was dropped at backward point by Jimmy Adams on 33 but fell when he chipped Liam Dawson back to the bowler.
With 40 minutes to go before lunch Hampshire finally received the second new ball and struck straight away – as Jonny Bairstow edged Kyle Abbott to third slip.
Yorkshire boast a 52-run lead with five wickets still intact and 68 overs left in the day.