This paper studies the asymptotic behavior of a one-dimensional directed polymer in a random medium. The latter is represented by a Gaussian field B H on R + × R with fractional Brownian behavior in time (Hurst parameter H ) and arbitrary function-valued behavior in space. The partition function of such a polymer isHere b is a continuous-time nearest neighbor random walk on Z with fixed intensity 2κ, defined on a complete probability space P b independent of B H . The spatial covariance structure of B H is assumed to be homogeneous and periodic with period 2π . For H < 1 2 , we prove existence and positivity of the Lyapunov exponent defined as the almost sure limit lim t→∞ t −1 log u(t). For H > 1 2 , we prove that the upper and lower almost sure limits lim sup t→∞ t −2H log u(t) and lim inf t→∞ (t −2H log t) log u(t) are non-trivial in the sense that they are bounded respectively above and below by finite, strictly positive constants. Thus, as H passes through 1 2 , the exponential behavior of u(t) changes abruptly. This can be considered as a phase transition phenomenon. Novel tools used in this paper include sub-Gaussian concentration theory via the Malliavin calculus, detailed analyses of the long-range memory of fractional Brownian motion, and an almost-superadditivity property.