  The MEG II experiment, based at the Paul Scherrer Institut in Switzerland,
reports the result of a search for the decay $\mu^+\to e^+\gamma$ from data
taken in the first physics run in 2021. No excess of events over the expected
background is observed, yielding an upper limit on the branching ratio of
B($\mu^+\to e^+\gamma$) < $7.5 \times 10^{-13}$ (90% C.L.). The combination of
this result and the limit obtained by MEG gives B($\mu^+\to e^+\gamma$) < $3.1
\times 10^{-13}$ (90% C.L.), which is the most stringent limit to date. A
ten-fold larger sample of data is being collected during the years 2022-2023,
and data-taking will continue in the coming years.
