  The dispersion measure of fast radio bursts (FRBs), arising from the
interactions of the pulses with free electrons along the propagation path,
constitutes a unique probe of the cosmic baryon distribution. Their
constraining power is further enhanced in combination with observations of the
foreground large-scale structure and intervening galaxies. In this work, we
present the first constraints on the partition of the cosmic baryons between
the intergalactic medium (IGM) and circumgalactic medium (CGM), inferred from
the FLIMFLAM spectroscopic survey. In its first data release, the FLIMFLAM
survey targeted galaxies in the foreground of 8 localized FRBs. Using Bayesian
techniques, we reconstruct the underlying ~Mpc-scale matter density field that
is traced by the IGM gas. Simultaneously, deeper spectroscopy of intervening
foreground galaxies (at impact parameters $b_\perp \lesssim r_{200}$) and the
FRB host galaxies constrains the contribution from the CGM. Applying Bayesian
parameter inference to our data and assuming a fiducial set of priors, we infer
the IGM cosmic baryon fraction to be $f_{\rm igm}=0.59^{+0.11}_{-0.10}$, and a
CGM gas fraction of $f_{\rm gas} = 0.55^{+0.26}_{-0.29}$ for $10^{10}\,M_\odot
\lesssim M_{\rm halo}\lesssim 10^{13}\,M_\odot$ halos. The mean FRB host
dispersion measure (rest-frame) in our sample is $\langle \rm{DM_{host}}\rangle
= 90^{+29}_{-19}\rm{pc~cm^{-3}}$, of which $\langle{\rm DM_{host}^{unk}}\rangle
=69^{+28}_{-19}~\rm{pc~cm^{-3}}$ arises from the host galaxy ISM and/or the FRB
progenitor environment. While our current $f_{\rm igm}$ and $f_{\rm gas}$
uncertainties are too broad to constrain most galactic feedback models, this
result marks the first measurement of the IGM and CGM baryon fractions, as well
as the first systematic separation of the FRB host dispersion measure into two
components: arising from the halo and from the inner ISM/FRB engine.
