I didn't even know milk allergy was a separate thing! But no, I can drink milk if it's the lactose-free kind (which is actually milk with added lactase), and the rash is also a symptom of intolerance.

Tests last year showed that I was sensitive to lactose rather than outright intolerant, but the pharmacist thinks that either I've worsened in the last 12 months, or a week of eating ice cream after almost no lactose at all meant I had an unusually strong reaction.