What happens if you freeze a soft serve of ice cream?
Soft serve ice cream is soft because:


It's warmer than hard ice cream. According to Wikipedia, it's generally made at about -4 C, rather than -15 C for hard ice cream. It's also served at a higher temperature -- 14 to 25 degrees F, compared to 5 to 7 degrees F for hard ice cream. 
It contains more air. The technical term for air introduced during freezing is overrun, and it's calculated based on the percent increase in volume of the ice cream. The more air, the fluffier and softer the mouth-feel. 
It usually contains less fat than hard ice cream, which is mitigated by the soft creamy mouth-feel resulting from the warmer temperature. 


If you were to freeze soft ice cream, you'd lose the advantage of item 1. In addition, your freezer takes longer to freeze ice cream than an ice cream maker, which may result in ice crystals in the ice cream. Furthermore, once the soft ice cream chills to the freezer temperature (usually around 0F), the lower fat content may decrease the perceived creaminess, because of the information in item 3. 

Of course, having said all of that, I haven't tried the experiment, so I'd be interested in hearing first hand stories.