  We present the results from a full polarization study carried out with ALMA
during the first VLBI campaign, which was conducted in Apr 2017 in the
$\lambda$3mm and $\lambda$1.3mm bands, in concert with the Global mm-VLBI Array
(GMVA) and the Event Horizon Telescope (EHT), respectively. We determine the
polarization and Faraday properties of all VLBI targets, including Sgr A*, M87,
and a dozen radio-loud AGN. We detect high linear polarization fractions
(2-15%) and large rotation measures (RM $>10^{3.3}-10^{5.5}$ rad m$^{-2}$). For
Sgr A* we report a mean RM of $(-4.2\pm0.3) \times10^5$ rad m$^{-2}$ at 1.3 mm,
consistent with measurements over the past decade, and, for the first time, an
RM of $(-2.1\pm0.1) \times10^5$ rad m$^{-2}$ at 3 mm, suggesting that about
half of the Faraday rotation at 1.3 mm may occur between the 3 mm photosphere
and the 1.3 mm source. We also report the first unambiguous measurement of RM
toward the M87 nucleus at mm wavelengths, which undergoes significant changes
in magnitude and sign reversals on a one year time-scale, spanning the range
from -1.2 to 0.3 $\times\,10^5$ rad m$^{-2}$ at 3 mm and -4.1 to 1.5
$\times\,10^5$ rad m$^{-2}$ at 1.3 mm. Given this time variability, we argue
that, unlike the case of Sgr A*, the RM in M87 does not provide an accurate
estimate of the mass accretion rate onto the black hole. We put forward a
two-component model, comprised of a variable compact region and a static
extended region, that can simultaneously explain the polarimetric properties
observed by both the EHT and ALMA. These measurements provide critical
constraints for the calibration, analysis, and interpretation of simultaneously
obtained VLBI data with the EHT and GMVA.
