Dating a girl that might have a boyfriend?

[Redeye]

Met this girl through some people at work. We went out a few times as a group. First night she mentioned she had a boyfriend. A week or so later she mentioned that they had split up after 2 years. I casually asked her to drinks a week after that. We ended up kissing. Started dating not to long after that. We've been dating for a month now.

She refers to him as her ex but still receives texts from him, and his facebook photo is still a pic of them.

I really like her so I don't want to jump to conclusions but what is the best way to make sure they still arent dating and that ive been lied to?
His Facebook is a picture of them. Not hers.

[illest]

Wow I would not be cool with my gf having a picture with her ex as her default. Obviously the only way to tell if she's with him is to secretly check her phone or follow her around and see what she does all day.