  In this paper we study the relation between the existence of a conformal
measure on the Julia set $J(f)$ of a transcendental meromorphic map $f$ and the
existence of zero of the topological pressure function $t \mapsto P(f, t)$ for
the map $f$. In particular, we show that if $f$ is hyperbolic and there exists
a $t$-conformal measure which is not totally supported on the set of escaping
points, then $P(f, t) = 0$. On the other hand, for a wide class of maps $f$,
including arbitrary maps with at most finitely many poles and finite set of
singular values and hyperbolic maps with at most finitely many poles and
bounded set of singular values, if $P(f, t) = 0$, we construct a $t$-conformal
measure on $J(f)$. This partially answers a question of R.D. Mauldin.
