Wikipedia concurs:

Any countable nonstandard model of arithmetic has order type ω + (ω* + ω) · η, where ω is the order type of the standard natural numbers, ω* is the dual order (an infinite decreasing sequence) and η is the order type of the rational numbers. In other words, a countable nonstandard model begins with an infinite increasing sequence (the standard elements of the model). This is followed by a collection of "blocks," each of order type ω* + ω, the order type of the integers. These blocks are in turn densely ordered with the order type of the rationals. The result follows fairly easily because it is easy to see that the non standard numbers have to be dense and linearly ordered without endpoints, and the rationals are the only countable dense linear order without endpoints.

Edit: Eliezer seems to have been aware of this, and gave a valid reply to your comment, so I won't call it a "mistake" anymore. I do think some rewording or a clarifying annotation within the OP would be helpful, though.