  We present a study of weak, thermonuclear X-ray bursts from the accreting
millisecond X-ray pulsar SAX J1808.4-3658. We focus on a burst observed with
the Neutron Star Interior Composition Explorer on 2019 August 9, and describe a
similar burst observed with the Rossi X-ray Timing Explorer in 2005 June. These
bursts occurred soon after outburst onset, $2.9$ and $1.1$ days, after the
first indications of fresh accretion. We measure peak burst bolometric fluxes
of $6.98 \pm 0.50 \times 10^{-9}$ and $1.54 \pm 0.10 \times 10^{-8}$ erg
cm$^{-2}$ s$^{-1}$, respectively, which are factors of $\approx 30$ and $15$
less than the peak flux of the brightest, helium-powered bursts observed from
this source. From spectral modeling we estimate accretion rates and accreted
columns at the time of each burst. For the 2019 burst we estimate an accretion
rate of $\dot M \approx 1.4-1.6 \times 10^{-10}$ $M_{\odot}$ yr$^{-1}$, and a
column in the range $3.9-5.1 \times 10^7$ g cm$^{-2}$. For the 2005 event the
accretion rate was similar, but the accreted column was half of that estimated
for the 2019 burst. The low accretion rates, modest columns, and evidence for a
cool neutron star in quiescence, suggest these bursts are triggered by
thermally unstable CNO cycle hydrogen-burning. The post-burst flux level in the
2019 event appears offset from the pre-burst level by an amount consistent with
quasi-stable hydrogen-burning due to the temperature-insensitive, hot-CNO
cycle, further suggesting hydrogen-burning as the primary fuel source. This
provides strong observational evidence for hydrogen-triggered bursts. We
discuss our results in the context of previous theoretical modeling.
