The Lyman-β forest refers to a region in the spectra of distant quasars that lies between the rest-frame Lyman-β and Lyman-γ emissions. The forest in this region is dominated by a combination of absorption due to resonant Lyα and Lyβ scattering. When considering the 1D Lyβ forest in addition to the 1D Lyα forest, the full statistical description of the data requires four 1D power spectra: Lyα and Lyβ auto-power spectra and the Lyα-Lyβ real and imaginary cross-power spectra. We describe how these can be measured using an optimal quadratic estimator that naturally disentangles Lyα and Lyβ contributions. Using a sample of approximately 60,000 quasar sight-lines from the BOSS Data Release 9, we make the measurement of the one-dimensional power spectrum of fluctuations due to the Lyβ resonant scattering. While we have not corrected our measurements for resolution damping of the power and other systematic effects carefully enough to use them for cosmological constraints, we can robustly conclude the following: i) Lyβ power spectrum and Lyα-Lyβ cross spectra are detected with high statistical significance; ii) the cross-correlation coefficient is ≈ 1 on large scales; iii) the Lyβ measurements are contaminated by the associated OVI absorption, which is analogous to the SiIII contamination of the Lyα forest. Measurements of the Lyβ forest will allow extension of the usable path-length for the Lyα measurements while allowing a better understanding of the physics of intergalactic medium and thus more robust cosmological constraints.