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We characterize a passively carrier envelope phase-stabilized Er:fiber laser frequency comb at 1550 nm. Phase noise of single comb lines over nearly an optical octave is measured via a delayed self-heterodyne beat of a cw clean-up laser. Repetition-rate stabilization achieves a RF-reference limited line-width. A lock to a low-noise optical reference shows Hz level out-of-loop line-widths. The measured phase noise is in excellent agreement with the elastic tape model with a fix point at zero frequency.