  We develop the local-global theory of blocks for profinite groups. Given a
field $k$ of characteristic $p$ and a profinite group $G$, one may express the
completed group algebra $k[[G]]$ as a product $\prod_{i\in I}B_i$ of closed
indecomposable algebras, called the blocks of $G$. To each block $B$ of $G$ we
associate a pro-$p$ subgroup of $G$, called the defect group of $B$, unique up
to conjugacy in $G$. We give several characterizations of the defect group in
analogy with defect groups of blocks of finite groups. Our main theorem is a
Brauer correspondence between the blocks of $G$ with defect group $D$ and the
blocks of the normalizer $N_G(D)$ with defect group $D$.
