This paper proposes a nonlinear attitude observer, built on the special orthogonal group SO(3), that resorts to body measurements of a single constant inertial vector, in addition to the angular velocity obtained from a set of three high-grade fiber optic gyros, which are herein assumed sensitive enough to measure the Earth's rotation vector. This strategy contrasts with typical attitude solutions that require measurements of two non-collinear inertial vectors, and can be very useful in scenarios with either strong magnetic anomalies or highly-accelerated vehicles, where either the magnetometer or accelerometer measurements may no longer be useful for attitude estimation in traditional solutions. The attitude estimation solution features a tuning parameter, in the form of a positive scalar gain, and the nonlinear error dynamics associated to the observer are shown to be locally exponentially stable. Simulation results with realistic noise are presented that allow to assess the performance of the proposed solution.