  The simplicial volume of oriented closed connected smooth manifolds that
admit a non-trivial smooth $S^1$-action vanishes. In the present work we prove
a version of this result for the integral foliated simplicial volume of
aspherical manifolds: The integral foliated simplicial volume of aspherical
oriented closed connected smooth manifolds that admit a non-trivial smooth
$S^1$-action vanishes. Our proof uses the geometric construction of Yano's
proof for ordinary simplicial volume as well as the parametrised uniform
boundary condition for $S^1$.
