The Oval (in Kennington, London) hosted the very first FA Cup Final, in 1872, and 19 more between 1874 and 1892. In 1880 it became the very first ground in England to host international Test cricket. As such it's actually a better answer to this question than Bramall Lane, which has only hosted one Test match (England v. Australia, in 1902) and the replay of an FA Cup Final (in 1912, when Barnsley beat West Bromwich Albion 1–0).
You could argue that The Oval is a cricket ground rather than a football ground. My response to that would be "And people call me pedantic?"
I suspect what the setter meant was "Which is the only current League football ground to have hosted both an FA Cup Final and a cricket Test match?"
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