Due to significant technological progress in the field of artificial intelligence, conversational agents have the potential to become smarter, deepen the interaction with their users, and overcome a function of merely assisting. Since humans often treat computers as social actors, theories on interpersonal relationships can be applied to human-machine interaction. Taking these theories into account in designing conversational agents provides the basis for a collaborative and benevolent long-term relationship, which can result in virtual companionship. However, we lack prescriptive design knowledge for virtual companionship. We addressed this with a systematic and iterative design science research approach, deriving meta-requirements and five theoretically grounded design principles. We evaluated our prescriptive design knowledge by taking a two-way approach, first instantiating and evaluating the virtual classmate Sarah, and second analyzing Replika, an existing virtual companion. Our results show that with virtual companionship, conversational agents can incorporate the construct of companionship known from human-human relationships by addressing the need to belong, to build interpersonal trust, social exchange, and a reciprocal and benevolent interaction. The findings are summarized in a nascent design theory for virtual companionship, providing guidance on how our design prescriptions can be instantiated and adapted to different domains and applications of conversational agents.