  We show that for a given holomorphic noncharacteristic surface S in
two-dimensional complex space, and a given holomorphic function on S, there
exists a unique meromorphic solution of Burgers' equation which blows up on S.
This proves the convergence of the formal Laurent series expansion found by the
Painlev\'e test. The method used is an adaptation of Nirenberg's iterative
proof of the abstract Cauchy-Kowalevski theorem.
