Speech enhancement attenuates interfering sounds in speech signals but may introduce artifacts that perceivably deteriorate the output signal. We propose a method for controlling the trade-off between the attenuation of the interfering background signal and the loss of sound quality. A deep neural network estimates the attenuation of the separated background signal such that the sound quality, quantified using the Artifact-related Perceptual Score, meets an adjustable target. Subjective evaluations indicate that consistent sound quality is obtained across various input signals. Our experiments show that the proposed method is able to control the trade-off with an accuracy that is adequate for real-world dialogue enhancement applications.