In this paper, we analyze the optimal default strategy of a firm when debt is convertible into equity. For this purpose, we consider a convertible consol bond in a time-independent model in the presence of bankruptcy costs and tax de-ductibility. The optimal default and conversion strategy result from a game be-tween equity and debt holders. A closed-form solution for the optimal default barrier exists if the firm pays no dividends. We show that an optimal default of convertible debt occurs earlier than a default of otherwise identical straight debt. A further comparison of the optimal default strategy with the strategy for straight debt shows that the value of convertible debt is lower when the firm follows the optimal strategy rather than the strategy for straight debt. Furthermore, we find that the important difference between the default barrier for convertible debt and identical but non-convertible debt rises with the conversion ratio, the coupon, a lower tax rate, and a lower payoff rate.