  In this paper, we provide an explicit construction of weight $0$ meromorphic
modular forms. Following work of Petersson, we build these via Poincar\'e
series. There are two main aspects of our investigation which differ from his
approach. Firstly, the naive definition of the Poincar\'e series diverges and
one must analytically continue via Hecke's trick. Hecke's trick is further
complicated in our situation by the fact that the Fourier expansion does not
converge everywhere due to singularities in the upper half-plane so it cannot
solely be used to analytically continue the functions. To explain the second
difference, we recall that Petersson constructed linear combinations from a
family of meromorphic functions which are modular if a certain principal parts
condition is satisfied. In contrast to this, we construct linear combinations
from a family of non-meromorphic modular forms, known as polar harmonic Maass
forms, which are meromorphic whenever the principal parts condition is
satisfied.
