A quantum theory of interacting particles with rigorous Lorentz covariance and rigorous conservation laws of Lee type is set up; the process A$\leftrightarrow$B$_1$+B$_2$ is permitted via a point interaction. For simplicity the particles are assumed to be scalars. The complete solution of the lowest `sector' of interest is straightforward in configuration space for the case of one space dimension. It is in the case of three space dimensions that the usual difficulties associated with a point interaction arise; it is shown that the use of a cut-off is not a reliable procedure in this context, and a section of the paper is devoted to the development of a device for dealing with these difficulties which differs from the usual renormalization method. It is found that applying this device to the covariant model leads to a theory which is without infinite renormalization of any kind, and which admits a legitimate `physical' interpretation after certain necessary `supplementary conditions' are applied. The S-states in the lowest sector are fully described in configuration space. Finally, an indication of the variety of different models which may be obtained is given, and it is found that certain models which are at first sight unacceptable on account of their structure in the absence of interaction lead in fact to `physical' consequences which are perfectly satisfactory.