People can visit the exhibition free of charge between 5pm and 8pm on Thursday and Friday, February 24 and 25, during half term week.
Visitors will also be able to enjoy a free cup of tea or coffee and biscuits.
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Pride of place in the exhibition goes to the rare and beautiful gold coin called a Half-Angel, which dates back to the reign of Richard III (1452-1485).
It was found within two miles of Buckingham by local metal detectorist Dave Bethell.
Old Gaol Museum trustee Mike Wisbach said: "Only seven Richard III Half-Angels exist.
"We are very lucky to have this local find - the only one of the seven to be on public display."
As well as the Half-Angel coin (which would have bought 40 gallons of ale back in the 1400s), visitors can see a complete suit of replica armour from the time of Richard III.
Mike said: "We are offering free entry to the exhibition to say a big thank you to the people of Buckingham for supporting the Old Gaol Museum over the years.
"The exhibition comes to an end on February 28, so it’s your last chance."
The exhibition is in the covered Old Exercise Yard of the Old Gaol building, accessed through the main door facing Buckingham market.
The Richard III Half-Angel Exhibition has been made possible by grants from the Rothschild Foundation, the V&A, the Art Fund and the Richard III Society.