  We study a matrix product representation of the Bethe ansatz state for the
Lieb-Linger model describing the one-dimensional Bose gas with delta-function
interaction. We first construct eigenstates of the discretized model in the
form of matrix product states using the algebraic Bethe ansatz. Continuous
matrix product states are then exactly obtained in the continuum limit with a
finite number of particles. The factorizing $F$-matrices in the lattice model
are indispensable for the continuous matrix product states and lead to a marked
reduction from the original bosonic system with infinite degrees of freedom to
the five-vertex model.
