We show that a graph of girth greater than $6 \log k + 3$ and minimum degree at least 3 has a minor of minimum degree greater than $k$. This is best possible up to a factor of at most $9/4$. As a corollary, every graph of girth at least $6\log r + 3\log\log r + c$ and minimum degree at least 3 has a $K_r$ minor.
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