We study the problem of transmitting a source sample with minimum distortion over an infinite-bandwidth additive white Gaussian noise channel under an energy constraint. To that end, we construct a joint source--channel coding scheme using analog pulse position modulation (PPM) and bound its quadratic distortion. We show that this scheme outperforms existing techniques since its quadratic distortion attains both the exponential and polynomial decay orders of Burnashev's outer bound. We supplement our theoretical results with numerical simulations and comparisons to existing schemes.