We study a commonly-used second-kind boundary-integral equation for solving the Helmholtz exterior Neumann problem at high frequency, where, writing Gamma for the boundary of the obstacle, the relevant integral operators map L^2(Gamma ) to itself. We prove new frequency-explicit bounds on the norms of both the integral operator and its inverse. The bounds on the norm are valid for piecewise-smooth Gamma and are sharp up to factors of log k (where k is the wavenumber), and the bounds on the norm of the inverse are valid for smooth Gamma and are observed to be sharp at least when Gamma is smooth with strictly-positive curvature. Together, these results give bounds on the condition number of the operator on L^2(Gamma ); this is the first time L^2(Gamma ) condition-number bounds have been proved for this operator for obstacles other than balls.