The recent announcement of the detection of gravitational waves by LIGO comes almost exactly 100 years after Einstein predicted gravitational waves in 1916. Are there other examples (in the era of modern science) of such a long delay between a theoretical prediction and an experimental confirmation?
There is inherent vagueness in dating "predictions" and "confirmations" in many cases. For instance, who predicted heliocentrism? Copernicus, Kepler, Newton, perhaps Aristarchus? When was it confirmed? Another difficulty, also present in this example, is with predictions across major shifts in science (which happened about every 100 years since 17th century). There is more or less clear sense of confirming a prediction formulated within a theory that stays in place for a period of time, as was the case with the detection of gravitational waves, but over longer periods of time it is not so clear. Clean cases of confirmation show a strong selection bias towards short periods of time, and this one is indeed unusual.
Dalton conjectured atomic bonds in forming chemical compounds in 1803 by analyzing known chemical reactions, development of chemistry throughout 19th century supported it indirectly, but many physicists (perhaps most) rejected their existence until early 20th century. When atoms were finally confirmed experimentally though they weren't the kinds of indivisible balls with hooks that Dalton imagined. Does it count?
Another curious example is Michell's prediction of black holes in 1784, he even anticipated how they might be detected:"If there should really exist in nature any bodies, whose density is not less than that of the sun, and whose diameters are more than 500 times the diameter of the sun, since their light could not arrive at us... we could have no information from light; yet, if any other luminous bodies should happen to revolve about them we might still perhaps from the motions of these revolving bodies infer the existence of the central ones with some degree of probability...". Of course, Michell was working with Newtonian gravity and corpuscular optics, and when black holes were finally found the governing theories were general relativity and quantum electrodynamics (with wave optics rising and falling in between). Should we count the prediction from Schwarzschild's 1916 paper instead? Dating confirmation is also tricky: Cygnus X-1 was discovered in 1964 and suspected to be a black hole since. But the consensus about that only started forming in 1970s, and even in 1980s its status was still somewhat controversial.
I think Guido makes a good point in a comment that the analogue in math would be proving an old conjecture, and for this there are many examples that were settled after over 100 years. Besides Fermat's Last Theorem there is:
the Poincaré conjecture, posed around 1900 and settled over 100 years later by Perelman,
the prime number theorem, conjectured independently by Legendre and Gauss in the 1790s, and proved independently by Hadamard and de la Valee Poussin in 1896,
the Gauss class number one problem, posed by Gauss around 1800 and settled in the second half of the 20th century independently by Heegner, Baker, and Stark, with further aspects settled later by work of Gross and Zagier,
the transcendence of e and pi, which was conjectured by Lambert in 1761 but proved in the 1870s and 1880s by Hermite and Lindemann.
Another old problem was the classification of constructible regular polygons. This was initiated by the ancient Greeks and settled 2000+ years later by Gauss, although the ancient Greeks did not predict which polygons were constructible (or for that matter if there were any constraints at all).
If you want to allow disproofs of a conjecture after a long time, Euler's conjecture from 1761 that for n > 1 a sum of n-th powers is not an n-th power in positive integers when we allow only fewer than n summands (and of course require at least two summands) was disproved for n = 4 and 5 by computer searches in the 20th century: Elkies for n = 4 and Lander and Parkin for n = 5.
It depends on what you call "modern" science. In one case, prediction was made in the antiquity, and confirmation had to wait for more than 2000 years: I mean "atomic theory" of Leukippe and Demokritus. The final confirmation was obtained only in 1905. (ON my opinion, 1905 is the "modern science). One can argue, of course whether this was a real scientific theory.
Democritus did not formulate his ideas on atomism as conjecture but rather as fact or theory, as do many researchers :-) In line with other editors' comments that the analogue in math would be proving an old conjecture, I would point out that Leibnizian theory of infinitesimals, assignable vs inassignable quantities, a more general relation of equality "up to", etc. was not "confirmed" until the 20th century with the work of Hewitt, Los, and Robinson. Thus the delay was almost as long as with Fermat's last theorem, and perhaps even longer if one takes into account that infinitesimal approaches go back to Kepler and Galileo (and of course even earlier to Archimedes).