  The thermodynamics of a magnetised Kerr-Newman black hole is studied to all
orders in the appended magnetic field $B$. The asymptotic properties of the
metric and other fields are dominated by the magnetic flux that extends to
infinity along the axis, leading to subtleties in the calculation of conserved
quantities such as the angular momentum and the mass. We present a detailed
discussion of the implementation of a Wald-type procedure to calculate the
angular momentum, showing how ambiguities that are absent in the usual
asymptotically-flat case may be resolved by the requirement of gauge
invariance. We also present a formalism from which we are able to obtain an
expression for the mass of the magnetised black holes. The expressions for the
mass and the angular momentum are shown to be compatible with the first law of
thermodynamics and a Smarr type relation. Allowing the appended magnetic field
$B$ to vary results in an extra term in the first law of the form $-\mu dB$
where $\mu$ is interpreted as an induced magnetic moment. Minimising the total
energy with respect to the total charge $Q$ at fixed values of the angular
momentum and energy of the seed metric allows an investigation of Wald's
process. The Meissner effect is shown to hold for electrically neutral extreme
black holes. We also present a derivation of the angular momentum for black
holes in the four-dimensional STU model, which is ${\cal N}=2$ supergravity
coupled to three vector multiplets.
