We report the discovery of a highly coherent oscillation in a type-I X-ray burst observed from 4U 1916-053 by the Rossi X-ray Timing Explorer (RXTE). The oscillation was most strongly detected approx. 1 s after the burst onset at a frequency of 269.4 Hz, and it increased in frequency over the following 4 seconds of the burst decay to a maximum of around 272 Hz. The total drift of 3.5 Hz represents the largest fractional change in frequency (1.3%) yet observed in any burst oscillation. If the asymptotic frequency of the oscillation is interpreted in terms of a decoupled surface burning layer, the implied neutron star spin period is around 3.7 ms. However, the expansion of the burning layer required to explain the 3.5 Hz drift during the burst is around 80 m, substantially larger than expected theoretically. The oscillation was not present in the persistent emission before the burst, nor in the initial rise. When detected its amplitude was 6-12% (RMS) with a roughly sinusoidal profile. The burst containing the oscillation showed no evidence for photospheric radius expansion, while at least 5 of the other 9 bursts observed from the source by RXTE during 1996 and 1998 did. No comparable oscillations were detected in the other bursts. A pair of kilohertz quasi-periodic oscillations (QPOs) has been previously reported from this source with a mean separation of 348 +/- 12 Hz. 4U 1916-053 is the first example of a source where the burst oscillation frequency is significantly smaller than the frequency separation of the kHz QPOs.