  A new timing detector measuring ~50 MeV/c positrons is under development for
the MEG II experiment, aiming at a time resolution $\sigma_t \sim
30~\mathrm{ps}$. The resolution is expected to be achieved by measuring each
positron time with multiple counters made of plastic scintillator readout by
silicon photomultipliers (SiPMs). The purpose of this work is to demonstrate
the time resolution for ~50 MeV/c positrons using prototype counters. Counters
with dimensions of $90\times 40\times 5~\mathrm{mm}^3$ readout by six SiPMs
(three on each $40\times 5~\mathrm{mm}^2$ plane) were built with SiPMs from
Hamamatsu Photonics and AdvanSiD and tested in a positron beam at the
DA$\Phi$NE Beam Test Facility. The time resolution was found to improve nearly
as the square root of the number of counter hits. A time resolution
$\sigma_t=26.2\pm1.3~\mathrm{ps}$ was obtained with eight counters with
Hamamatsu SiPMs. These results suggest that the design resolution is achievable
in the MEG II experiment.
