The speckle pattern produced when a laser is scattered by a disordered medium has recently been shown to give a surprisingly accurate or broadband measurement of wavelength. Here we show that deep learning is an ideal approach to analyse wavelength variations using a speckle wavemeter due to its ability to identify trends and overcome low signal to noise ratio in complex datasets. This combination enables wavelength measurement at high resolution and over a broad operating range in a single step, which has not been possible with previous approaches. We demonstrate attometre-scale wavelength resolution over an operating range from 488 nm to 976 nm. This dynamic range is six orders of magnitude beyond the state of the art.