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It is well known that all Borel subgroups of a linear algebraic group are conjugate. Berest, Eshmatov, and Eshmatov have shown that this result also holds for the automorphism group Aut ⁢ (픸 2) of the affine plane. In this paper, we describe all Borel subgroups of the complex Cremona group Bir ⁢ (ℙ 2) up to conjugation, proving in particular that they are not necessarily conjugate. In principle, this fact answers a question of Popov. More precisely, we prove that Bir ⁢ (ℙ 2) admits Borel subgroups of any rank r ∈ { 0 , 1 , 2 } and that all Borel subgroups of rank r ∈ { 1 , 2 } are conjugate. In rank 0, there is a one-to-one correspondence between conjugacy classes of Borel subgroups of rank 0 and hyperelliptic curves of genus ℊ ≥ 1 . Hence, the conjugacy class of a rank 0 Borel subgroup admits two invariants: a discrete one, the genus ℊ , and a continuous one, corresponding to the coarse moduli space of hyperelliptic curves of genus ℊ . This moduli space is of dimension 2 ⁢ ℊ - 1 . [ABSTRACT FROM AUTHOR]