Henrik, the discrete Fourier transform works for any signal, not only for ones with the first and last elements equal.
We can check it formally, making sure that the explicit inverse Fourier transform
Table[Chop[1/Sqrt[n] (Fourier[data] . Table[Exp[-2 \[Pi] I t s / n ], {s, 0, n-1}])], {t, 0, n-1}]
brings us back to the initial data. And this expression, in the case of my signal, includes Exp[-2 [Pi] I t * 0.25 ] rather than Exp[-2 [Pi] I t * 0.255102 ], so the true frequency must be 0.25.
