This is a poorly worded statement, but it is indeed true. Ordinarily, the statement is,
"An object does not accelerate unless a net force acts on it."
Here I use "unless" instead of "until" because the meaning is clearer, and "net" instead of "resultant" because it's more common terminology.
This statement is equivalent to saying,
"If there's a net force acting on an object, then it will accelerate."
Butchering this statement a bit, but retaining its meaning, we get:
"If there's a net force acting on an object, then it will continue to accelerate unless that net force is zero."
Omitting the first clause (because the author apparently assumed that you knew that there was a net force being applied to the object by default), we get the author's statement:
"An object will continue to accelerate unless the net force on it is zero."