We study an optimal investment problem under incomplete information for an investor with constant relative risk aversion. We assume that the investor can only observe asset prices, but not the instantaneous returns. Furthermore, we assume that the instatantaneous returns follow an Ornstein--Uhlenbeck process, and that their initial distribution is Gaussian. We analytically solve the Bellman equation for this problem, and indentify the optimal investment strategy under incomplete information. We explore the relationship between the value function under partial observation and the value function under full observation, and derive a formula for the economic value of information. Furthermore, we discuss how the optimal strategy under partial observation can be computed from the optimal strategy for an investor with full observation. Explicit solutions are presented in a model with only one risky asset.