The short answer is "No." An unequivocal, blunt, clear, no.
Here's why. The Product Owner represents the customer (and possibly internal stakeholders). The PO creates the product backlog and the iteration backlog. The PO accepts stories (each with their definition of done). The PO represents the outside world to the team.
The Scrum Master is a facilitative project manager, who focuses on the team's process and risks. The SM helps the team by removing impediments or escalating those impediments. The SM makes sure the team performs the Scrum rituals/ceremonies for the benefit of the team.
For example, the SM makes sure the team performs a demo at the end of an iteration. This helps the team prepare to assess their work in a retrospective. The SM makes sure the team retrospects on their process and the product as it stands.
The SM focuses on creating and keeping the team environment so that the team can succeed at its work.
Especially when a team transitions to Scrum, both positions are full time. If one person tried to do both, I don't see how one person could do all the work.
The PO position is a full-time job. The SM position is a full-time job. They each have a different focus.