Optical atomic clocks outperform the microwave clocks at the heart of current timing and navigation infrastructure by orders-of-magnitude, and recent progress in on-chip integration provides the opportunity to widely deploy them. Conventionally, a narrow linewidth laser interrogates the atomic transition, and an optical frequency comb divides its stabilised frequency (hundreds of terahertz) down to a microwave that is electronically countable. The laser(s) impose a significant size, weight, power and cost penalty, which is a current barrier to low-cost, scalable, fully on-chip optical clocks. Here, we position the frequency comb as the central engine of the clock by using Brillouin-amplified comb lines for atomic spectroscopy, which eliminates the need for stable clock lasers. Amplified comb lines reach an ultra-low frequency noise floor of 6 Hz²/Hz at 30 kHz offset, an order-of-magnitude below a high-end commercial fibre laser. We realize a clock based on the 5S_(1/2)→5D_(5/2) transition in ⁸⁷Rb with competitive fractional frequency stability of 2×10⁻¹³ at τ = 1 s, averaging to 9×10⁻¹⁵ at τ = 1000 s. Lastly, we demonstrate a photonic chip-integrated Brillouin amplifier with a record 62.9 dB gain and up to 40 dB transferred noise suppression when pumped with a simple, low-cost diode laser. This opens a new route to compact optical clocks and enables use of atomic transitions lacking low-noise lasers by broadening a comb to the required wavelength and amplifying single lines.