  The ground state hyperfine splitting of positronium, $\Delta_{\mathrm{HFS}}$,
is sensitive to high order corrections of QED. A new calculation up to
$\mathrm{O}(\alpha ^3 \ln \alpha)$ has revealed a $3.9\sigma$ discrepancy
between the QED prediction and the experimental results. This discrepancy might
either be due to systematic problems in the previous experiments or to
contributions beyond the Standard Model. We propose an experiment to measure
$\Delta_{\mathrm{HFS}}$ employing new methods designed to remedy the systematic
errors which may have affected the previous experiments. Our experiment will
provide an independent check of the discrepancy. The prototype run has been
finished and a result of $\Delta_{\mathrm{HFS}} = 203.380 4 \pm 0.008 4
\mathrm{GHz} (41 \mathrm{ppm})$ has been obtained. A measurement with a
precision of O(ppm) is expected within a few years.
