Let $X$ be a linearly ordered topological space with a countable dense subset. Does it necessarily follow that $X$ is metrizable?

A linearly ordered space is a set $X$ with a total order such that the open intervals form a basis for $X$. For example, every ordinal can be given the order topology and the usual Euclidean topology on the real line is the order topology. See the Order topology on Wikipedia. I am not asking for an example of a separable space that is not metrizable - there are many examples for that (there's no need for the Sorgenfrey line, you can even pick the trivial topology...)