  We introduce a new technique to constrain the line-of-sight integrated
electron density of our Galactic halo $\text{DM}_\text{MW,halo}$ through
analysis of the observed dispersion measure distributions of pulsars
$\text{DM}_\text{pulsar}$ and fast radio bursts $\text{DM}_\text{FRB}$. We
model these distributions, correcting for the Galactic interstellar medium,
with kernel density estimation---well-suited to the small data regime---to find
lower/upper bounds to the corrected
$\text{DM}_\text{pulsar}$/$\text{DM}_\text{FRB}$ distributions:
$\max[\text{DM}_\text{pulsar}] \approx 7\pm2 \text{ (stat)} \pm 9 \text{ (sys)
pc cm}^{-3}$ and $\min[\text{DM}_\text{FRB}] \approx 63^{+27}_{-21} \text{
(stat)} \pm 9 \text{ (sys) pc cm}^{-3}$. Using bootstrap resampling to estimate
uncertainties, we set conservative limits on the Galactic halo dispersion
measure $-2 < \text{DM}_\text{MW,halo} < 123 \text{pc cm}^{-3}$ (95\% c.l.).
The upper limit is especially conservative because it may include a
non-negligible contribution from the FRB host galaxies and a non-zero
contribution from the cosmic web. It strongly disfavors models where the Galaxy
has retained the majority of its baryons with a density profile tracking the
presumed dark matter density profile. Last, we perform Monte Carlo simulations
of larger FRB samples to validate our technique and assess the sensitivity of
ongoing and future surveys. We recover bounds of several tens $\text{pc
cm}^{-3}$ which may be sufficient to test whether the Galaxy has retained a
majority of its baryonic mass. We estimate that a sample of several thousand
FRBs will significantly tighten constraints on $\text{DM}_\text{MW,halo}$ and
offer a valuable complement to other analyses.
