With the proportional counter array on board the Rossi X-ray Timing Explore
r, we report the discovery of a decrease in the frequency of X-ray brightne
ss oscillations in the cooling tail of an X-ray burst from 4U 1636-53. This
is the first direct evidence for a spin-down of the pulsations seen during
thermonuclear bursts. We find that the spin-down episode is correlated wit
h the appearance in this bm:st of an extended tail of emission with a decay
timescale much longer than is seen in other bursts from 4U 1636-53 in the
same set of observations. We present both time-resolved energy and variabil
ity spectra during this burst and compare them with results from a second b
urst that shows neither a spin-down episode nor an extended tail. A spectra
l evolution study of the "spin-down" burst reveals a secondary signature of
weak radius expansion, which is not seen in other bursts and is correlated
with the spin-down episode; this may indicate a secondary thermonuclear en
ergy release. We interpret the spin-down episode in the context of an angul
ar momentum-conserving shell, which is reexpanded and therefore spun down b
y an additional thermonuclear energy release that could also explain the ex
tended X-ray tail.