  The classical Bj\"orling problem is to find the minimal surface containing a
given real analytic curve with tangent planes prescribed along the curve. We
consider the generalization of this problem to non-minimal constant mean
curvature (CMC) surfaces, and show that it can be solved via the loop group
formulation for such surfaces. The main result gives a way to compute the
holomorphic potential for the solution directly from the Bj\"orling data, using
only elementary differentiation, integration and holomorphic extensions of real
analytic functions. Combined with an Iwasawa decomposition of the loop group,
this gives the solution, in analogue to Schwarz's formula for the minimal case.
Some preliminary examples of applications to the construction of CMC surfaces
with special properties are given.
