Well, the statement was "their body only requires 1000 calories", not "their BMR is 1000 calories".
So yes, this person would hypothetically gain weight, more slowly than they had been at 2,000 calories.
HOWEVER, this hypothetical person can't exist. Nobody has a BMR so low that by the time you include daily activity (and let's assume no exercise at all) they're only burning 1000 calories per day. Nobody's that small. Maybe actual Little People, but not your average healthy adult.
So the figures are unrealistic, but in terms of understanding the math, yes she'd gain.
This is why it bugs me when I see things like "Eat 500 calories less than you were before to lose a pound a week". Um no. If you were gaining 2 pounds a week, you'd still gain 1 pound a week by eating 500 calories less. You need to eat less than *you need* not less than *you were*.
To be more realistic, take a person who was eating 2,000 calories and maintaining their overweight frame.
If their BMR is 1600 and they did on average 700 calories per week in exercise, they would daily burn 2000 calories. That's why they're maintaining weight right now.
But if they up the exercise to 1400 calories weekly plus eat 1600 calories instead of 2000, they are now creating a 500 calorie daily deficit (400 less calories in, plus 100 extra calories out) and would lose a pound a week. On average.
Deb, in New Zealand