  The notion of almost Gorenstein ring given by Barucci and Fr{\"o}berg
\cite{BF} in the case where the local rings are analytically unramified is
generalized, so that it works well also in the case where the rings are
analytically ramified. As a sequel, the problem of when the endomorphism
algebra $\m : \m$ of $\m$ is a Gorenstein ring is solved in full generality,
where $\m$ denotes the maximal ideal in a given Cohen-Macaulay local ring of
dimension one. Characterizations of almost Gorenstein rings are given in
connection with the principle of idealization. Examples are explored.
