We present the detection of a variable star with an $18.05$ minute period in far-ultraviolet (FUV) images of the globular cluster NGC 1851 taken with the Hubble Space Telescope (HST). A candidate optical counterpart lies on the red horizontal branch or the asymptotic giant branch star of the cluster, but it is statistically possible that this is a chance superposition. This interpretation is supported by optical spectroscopt obtained with HST/STIS: the spectrum contains none of the strong emission lines that would be expected if the object was a symbiotic star (i.e. a compact accretor fed by a giant donor). We therefore consider two other possibilities for the nature of FUV variable: (i) an intermediate polar (i.e. a compact binary containing an accreting magnetic white dwarf), or (ii) an AM CVn star (i.e. an interacting double-degenerate system). In the intermediate polar scenario, the object is expected to be an X-ray source. However, no X-rays are detected at its location in $\simeq 65$~ksec of {\em Chandra} imaging, which limits the X-ray luminosity to $L_X \leqslant 10^{32}$~erg~s$^{-1}$. We therefore favour the AM CVn interpretation, but a FUV spectrum is needed to distinguish conclusively between the two possibilities. If the object is an AM CVn binary, it would be the first such system known in any globular cluster.
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