Quantum field theories with exact but spontaneously broken conformal invariance have an intriguing feature: the vacuum energy (cosmological constant) in them is equal to zero. Up to now, the only known ultraviolet complete theories where conformal symmetry can be spontaneously broken were associated with supersymmetry (SUSY), with the most prominent example being the $\mathcal N$=4 SUSY Yang-Mills. In this Letter we show that the recently proposed biscalar conformal "fishnet" theory supports at the classical level a rich set of flat directions (moduli) along which conformal symmetry is spontaneously broken. We demonstrate that some of these vacua survive in the full quantum theory (in the planar limit, at the leading order of $1/N_c$ expansion) without any fine tuning. The vacuum energy is equal to zero along these flat directions, providing a first non-SUSY example of a four-dimensional quantum field theory with "natural" breaking of conformal symmetry.