We observed the Rapid Burster with Chandra when it was in the 'banana' state that usually precedes the type-II X-ray bursting 'island' state for which the source is particularly known. We employed the High-Energy Transmission Grating Spectrometer in combination with the ACIS-S detector in continuous clocking mode. The observation yielded 20 thermonuclear type-I X-ray bursts emitted from the neutron star surface with recurrence times between 0.9 and 1.2 hr, and an e-folding decay time scale of 1 min. We searched for narrow spectral features in the burst emission that could constrain the composition of the ashes of the nuclear burning and the compactness of the neutron star, but found none. The upper limit on narrow absorption lines between 2 and 6 keV is between 5 and 20 eV (single trial 3 sigma confidence level) and on absorption edges between 150 and 400 eV. The latter numbers are comparable to the levels predicted by Weinberg, Bildsten & Schatz (2006) for Eddington-limited thermonuclear bursts.