  It is shown that for a non-singular conservative shift on a topologically
mixing Markov subshift with Doeblin Condition the only possible absolutely
continuous shift-invariant measure is a Markov measure. Moreover, if it is not
equivalent to a homogeneous Markov measure then the shift is of Krieger-type
$\mathrm{III}_1$. A criterion for equivalence of Markov measures is included.
