  Given $\a \in [0,1]$ and $\varphi: \T \to \R$ measurable, the {\it
cylindircal cascade} $S_{\a,\varphi}$ is the map from $\T \times \R$ to itself
given by $S_{\a,\varphi} (x,y) = (x+\a,y+\varphi(x))$ that naturally appears in
the study of some ordinary differential equations on $\R^3$. In this paper, we
prove that for a set of full Lebesgue measure of $\a \in [0,1]$ the cylindrical
cascades $S_{\a,\varphi}$ are ergodic for every smooth function $\varphi$ with
a logarithmic singularity, provided that the average of $\varphi$ vanishes.
  Closely related to $S_{\a,\varphi}$ are the special flows constructed above
$R_\a$ and under $\varphi+c$ where $c \in \R$ is such that $\varphi+c>0$. In
the case of a function $\varphi$ with an asymmetric logarithmic singularity our
result gives the first examples of ergodic cascades $S_{\a,\varphi}$ with the
corresponding special flows being mixing.
  Indeed, when the latter flows are mixing the usual techniques used to prove
the {\it essential value criterion} for $S_{\a,\varphi}$, that is equivalent to
ergodicity, fail and we device a new method to prove this criterion that we
hope could be useful in tackling other problems of ergodicity for cocycles
preserving an infinite measure.
