  We measure the effective optical depth of HeII Ly\alpha\ absorption
\tau$_\mathsf{eff,HeII}$ at 2.3<z<3.5 in 17 UV-transmitting quasars observed
with UV spectrographs on the Hubble Space Telescope (HST). The median
\tau$_\mathsf{eff,HeII}$ values increase gradually from 1.95 at z=2.7 to 5.17
at z=3.4, but with a strong sightline-to-sightline variance. Many $\simeq$35
comoving Mpc regions of the z>3 intergalactic medium (IGM) remain transmissive
(\tau$_\mathsf{eff,HeII}$<4), and the gradual trend with redshift appears
consistent with density evolution of a fully reionized IGM. These modest
optical depths imply average HeII fractions of x$_\mathsf{HeII}$<0.01 and HeII
ionizing photon mean free paths of $\simeq$50 comoving Mpc at z$\simeq$3.4,
thus requiring that a substantial volume of the helium in the Universe was
already doubly ionized at early times; this stands in conflict with current
models of HeII reionization driven by luminous quasars. Along 10 sightlines we
measure the coeval HI Ly\alpha\ effective optical depths, allowing us to study
the density dependence of \tau$_\mathsf{eff,HeII}$ at z$\sim$3. We establish
that the dependence of \tau$_\mathsf{eff,HeII}$ on increasing
\tau$_\mathsf{eff,HI}$ is significantly shallower than expected from simple
models of an IGM reionized in HeII. This requires higher HeII photoionization
rates in overdense regions or underdense regions being not in photoionization
equilibrium. Moreover, there are very large fluctuations in
\tau$_\mathsf{eff,HeII}$ at all $\tau_\mathsf{eff,HI}$, which greatly exceed
the expectations from these simple models. These data present a distinct
challenge to scenarios of HeII reionization - an IGM where HeII appears to be
predominantly ionized at z$\simeq$3.4, and with a radiation field strength that
may be correlated with the density field, but exhibits large fluctuations at
all densities.
